The Graver Complexity of Integer Programming

In this article we establish an exponential lower bound on the Graver complexity of integer programs. This provides new type of evidence supporting the presumable intractability of integer programming. Specifically, we show that the Graver complexity of the incidence matrix of the complete bipartite graph K ₃,m satisfies g(m) = Ω(2 ^m ), with g(m) ≥ 17·2 ^m−3 –7 for every m > 3.     

DFT Calculations and Synthesis Reveal: Key Intermediates,Omitted Mechanisms,and Unsymmetrical Bimane Products

Theoretical and experimental mixed approaches are complementary and valuable. Our DFT calculations support the mechanism suggested by Kosower, adding to it a key diaziridine intermediate that determines the relative product distribution of this reaction. Our results are consistent with the formation of the diazoketene intermediate as the rate-limiting step. Based on curve fittings, first or second-order kinetics cannot be ruled out. This may indicate that more than one mechanism is simultaneously at play in this transformation. This unexpected outcome led us to study an alternative cyclopropenone intermediate. Although cyclopropenone is not likely to be formed under thermal conditions, adding it to the reaction mixture results in bimane structures. The most staggering finding from this investigation was the unanticipated generation of the unsymmetrical anti-(Me,Me)(Ph,Ph)bimane. The optimization of this route towards unsymmetrical bimanes will require additional investigation.     

Dual Role of Silicon-based Matrices in Electron Exchange Matrices for Waste Treatment

Para chloro aniline (PCA) is a common toxic pollutant found in pharmaceutical wastewater. Our study suggests a novel PCA treatment method based on a heterogeneous advanced oxidation process (AOP) that proceeds in an electron exchange matrix (EEM) prepared by the incorporation of redox-active specie in silica matrices using the sol-gel synthesis route. The results, which are supported by DFT calculations, show that the silicon skeleton of the EEM has two important roles, both as a porous matrix that hosts the redox species and as an oxidant species involved in the AOP. The calculations indicate that the formation of a radical on the nitrogen is favored. The suggested mechanism could shed light on the AOP, which proceeds in a heterogenous system, and on its application inside the understudied EEMs that, until now, have been a virtual black box. A better understanding of the mechanism could lead to improved control over the heterogeneous processes that can play a critical role in industries with the need to treat small amounts of toxic compounds at low concentrations, such as in the pharmaceutical industry.