A model of students’ combinatorial thinking

Combinatorial topics have become increasingly prevalent in K-12 and undergraduate curricula, yet research on combinatorics education indicates that students face difficulties when solving counting problems. The research community has not yet addressed students’ ways of thinking at a level that facilitates deeper understanding of how students conceptualize counting problems. To this end, a model of students’ combinatorial thinking was empirically and theoretically developed; it represents a conceptual analysis of students’ thinking related to counting and has been refined through analyzing students’ counting activity. In this paper, the model is presented, and relationships between formulas/expressions, counting processes, and sets of outcomes are elaborated. Additionally, the usefulness and potential explanatory power of the model are demonstrated through examining data both from a study the author conducted, and from existing literature on combinatorics education.     

Synthesis and Properties of Higher Nuclearity Polyazanes

Polyazanes (i.e., higher nuclearity homologues of hydrazines) with increasing numbers of bound nitrogen atoms (from 3 to 5), including the first pentazane ever described, were prepared by the addition of lower-order polyazanes to diazo reagents. A structure was obtained. It was shown that the polynitrogen chains adopt a helical conformation. DFT modeling shows that the arrangement persists in solution. Although the polyazanes are all reducing agents, they become less so as the number of nitrogens increases.     

Modified [(IPr)Pd(R‐acac)Cl] Complexes: Influence of the acac Substitution on the Catalytic Activity in Aryl Amination

The synthesis of a series of [(IPr)Pd(R‐acac)Cl] precatalysts (acac=acetylacetonato; IPr=1,3‐bis(2,6‐diisopropylphenyl)imidazol‐2‐ylidene), where the acac ligand on palladium has been systematically modified through terminal substitution, is reported. The following substituted acac ligands are employed in this study: dibenzoylmethanato (dbm), benzoylacetonato (bac), tetramethylheptanedionato (tmhd), and hexafluoroacetylacetonato (hfac). Full spectroscopic characterization of the new complexes is provided along with X‐ray studies for three of these. Investigation of their catalytic activity in cross‐coupling is also presented through a comparative study in an aryl amination reaction. The catalytic results show a strong correlation between the increased steric bulk of the acac substituents and an increased activation rate of the precatalyst, going from the acac to the tmhd ligand. This observation, along with the inertness of the hfac compound, seems to support our previous proposal for the activation mechanism of these complexes under cross‐coupling conditions.     

Cephem-Pyrazinoic Acid Conjugates: Circumventing Resistance in Mycobacterium tuberculosis

Tuberculosis (TB) is a leading source of infectious disease mortality globally. Antibiotic-resistant strains comprise an estimated 10 % of new TB cases and present an urgent need for novel therapeutics. β-lactam antibiotics have traditionally been ineffective against M. tuberculosis (Mtb), the causative agent of TB, due to the organism's inherent expression of β-lactamases that destroy the electrophilic β-lactam warhead. We have developed novel β-lactam conjugates, which exploit this inherent β-lactamase activity to achieve selective release of pyrazinoic acid (POA), the active form of a first-line TB drug. These conjugates are selectively active against M. tuberculosis and related mycobacteria, and activity is retained or even potentiated in multiple resistant strains and models. Preliminary mechanistic investigations suggest that both the POA “warhead” as well as the β-lactam “promoiety” contribute to the observed activity, demonstrating a codrug strategy with important implications for future TB therapy.     

Synthesis and Structures of Two Functionalized Isobenzofurans: (3aR,7aR,4S,5S)-1,3,3a,4,5,7a-Hexahydro-5-methyl-3-oxoisobenzofuran-4-carboxylic acid and (3aα,4β,7β,7aα)-4-[(Acetyloxy)methyl]-3a,4,7,7a-tetrahydro-7-methyl-1,3-isobenzofurandione

Abstract  

The title compounds, C₁₀H₁₂O₄ and C₁₂H₁₄O₅, were prepared as part of an ongoing study to explore the practical aspects of solventless reaction methods. As confirmed by the crystal structures, treatment of maleic anhydride with (2E,4E)-hexa-2,4-dien-1-ol or its acetate counterpart resulted in cis bicyclo [4.3.0] molecular frameworks. The racemic product derived from the dienol synthon crystallized in chiral space group P2₁2₁2₁ via a pasteurian resolution process with molecules organized by carboxyl⋯carboxyl and C–H⋯O contacts. The acetate derivative crystallizes in space group Cc.     

Complete 1H- and 13C NMR assignments of saponins from roots of Gypsophila trichotoma Wend

The assignments of 1H and 13C NMR spectra of two new aminoacyl triterpene saponins from roots of Gypsophila trichotoma Wend. are reported. In addition to 1D NMR methods, 2D NMR techniques (COSY, TOCSY, ROESY, HSQC, HMBC, and HSQC-TOCSY) were used for the assignments. The structures were completed by analysis of HR-ESI-MS and ESI-MS(n).     

Updating Network Flows Given Multiple,Heterogeneous Arc Attribute Changes

A dual ascent reoptimization technique is proposed for updating optimal flows for the minimum cost network flow problem (MCNFP) given any number of simultaneous, heterogeneous changes to the network attributes (i.e. supply at nodes, arc costs and arc capacities) and the optimal solutions to the prior primal and dual problems. Significant savings in computation time can be achieved through the use of reoptimization in place of solving a new MCNFP from scratch as each new problem instance (i.e. set of network attribute updates) arises. The proposed technique can be implemented with polynomial worst-case computational complexity. Extensive numerical experiments were designed and conducted to assess the computational benefits of employing the proposed reoptimization technique as compared with solution from scratch using comparable classic implementations of the original algorithms. This work was motivated by the need for the real-time provision of evacuation instructions to people seeking quick egress from a large sensor-equipped building that has come under attack by natural or terrorist forces, but has broad applicability.