Solution of sparse quasi-square rectangular systems by Gaussian elimination

We present a general method for the linear least-squares solutionof overdetermined and underdetermined systems. The method isparticularly efficient when the coefficient matrix is quasi-square,that is when the number of rows and number of columns is almostthe same. The numerical methods for linear least-squares problemsand minimum-norm solutions do not generally take account ofthis special characteristic. The proposed method is based onLU factorization of the original quasi-square matrix A, assumingthat A has full rank. In the overdetermined case, the LU factorsare used to compute a basis for the null space of A^T. The right-handside vector b is then projected onto this subspace and the least-squaressolution is obtained from the solution of this reduced problem.In the case of underdetermined systems, the desired solutionis again obtained through the solution of a reduced system.The use of this method may lead to important savings in computationaltime for both dense and sparse matrices. It is also shown inthe paper that, even in cases where the matrices are quite small,sparse solvers perform better than dense solvers. Some practicalexamples that illustrate the use of the method are included.     

Synthesis of Functionalized Glutamine‐ and Asparagine‐Type Peptoids – Scope and Limitations

N‐Alkylated glycine oligomers (‘peptoids’) can serve as potent peptidomimetic systems. Installing different functional groups can often be a challenge, and minimizes yields and purities. Here, we describe the synthesis of different amide‐containing submonomers which were obtained as free bases, as well as their incorporation into peptoids. By using the free amines, the coupling results on solid support could be improved, and various functionalized peptoids were prepared. Additionally, an interesting dimerization side reaction leading to cross‐linked peptoids was observed during synthesis.     

In Operando Analysis of Diffusion in Porous Metal-Organic Framework Catalysts

The potential to exert atomistic control over the structure of site-isolated catalyst sites, as well as the topology and chemical environment of interstitial pore spaces, has inspired efforts to apply porous metal-organic frameworks (MOFs) as catalysts for fine chemical synthesis. In analogy to enzyme-catalyzed reactions, MOF catalysts have been proposed as platforms in which substrate confinement could be used to achieve chemo- and stereoselectivities that are orthogonal to solution-phase catalysts. In order to leverage the tunable pore topology of MOFs to impact catalyst selectivity, catalysis must proceed at interstitial catalyst sites, rather than at solvent-exposed interfacial sites. This Minireview addresses challenges inherent to interstitial MOF catalysis by 1) describing the diffusional processes available to sorbates in porous materials, 2) discussing critical factors that impact the diffusion rate of substrates in porous materials, and 3) presenting in operando experimental strategies to assess the relative rates of substrate diffusion and catalyst turnover in MOF catalysis. It is anticipated that the continued development of in operando tools to evaluate substrate diffusion in porous catalysts will advance the application of these materials in fine chemical synthesis.