Probably certifiably correct <Emphasis Type="Italic">k</Emphasis>-means clustering

Recently, Bandeira (C R Math, 2015) introduced a new type of algorithm (the so-called probably certifiably correct algorithm) that combines fast solvers with the optimality certificates provided by convex relaxations. In this paper, we devise such an algorithm for the problem of k-means clustering. First, we prove that Peng and Wei’s semidefinite relaxation of k-means Peng and Wei (SIAM J Optim 18(1):186–205, 2007) is tight with high probability under a distribution of planted clusters called the stochastic ball model. Our proof follows from a new dual certificate for integral solutions of this semidefinite program. Next, we show how to test the optimality of a proposed k-means solution using this dual certificate in quasilinear time. Finally, we analyze a version of spectral clustering from Peng and Wei (SIAM J Optim 18(1):186–205, 2007) that is designed to solve k-means in the case of two clusters. In particular, we show that this quasilinear-time method typically recovers planted clusters under the stochastic ball model.     

Dynamics of chiral liquid crystals under applied shear

We simulate the alignment dynamics of cholesteric (chiral) rod-like liquid crystals by using a Landau-de Gennes (LdG) expression for microstructure evolution in response to flow. This study is motivated by recent advances in novel cholesteric nanorod dispersions. Prior work on the modelling of cholesterics has suffered from the restriction of helicity to only a single direction, often with a pre-imposed pitch, due to numerical difficulties. This has severely limited cholesteric modelling in regard to both accuracy and experimental relevance. Our simulations avoid this limitation. Relevant forces on rods include solvent-rod drag, nematic alignment, microstructure elasticity and chiral twist. Phase diagrams are developed to demonstrate the response of these systems to variations in chiral and flow forces. Our results indicate that for low shear rates, chiral and elastic forces prevent the rods from moving in response to flow. At high shear rates, the rods tumble and form unique transient structures (combinations of tumbling and cholesteric phases) as flow forces and chiral forces compete. Even if slight alignment is induced at the boundaries, the phase diagram substantially changes, chiefly by constraining the possible chiral phases. This work has immediate relevance to applications which exploit the optical properties of films solidified from cholesteric dispersions.     

Effect of direct current dielectrophoresis on the trajectory of a non-conducting colloidal sphere in a bent pore

Dielectrophoresis has shown a wide range of applications in microfluidic devices. Force approximations utilizing the point-dipole method in dielectrophoresis have provided convenient predictions for particle motion by neglecting interactions between the particle and its surrounding electric and flow fields. The validity of this approach, however, is unclear when the particle size is comparable to the characteristic length of the channel and when the particle is in close proximity to the channel wall. To address this issue, we apply an accurate numerical approach based on the boundary-element method (BEM) to solve the coupled electric field, flow, and particle motion. This method can handle much closer particle-wall distances than the other numerical approaches such as the finite-element method. Using the BEM and integrating the Maxwell stress tensor, we simulate an electrokinetic, spherical particle moving within a bent cylindrical pore to investigate how the dielectrophoretic force affects the particle's trajectory. In the simulation, both the particle and the channel wall are non-conducting, and the electric double layers adjacent to the solid surfaces are assumed to be thin with respect to the particle radius and particle-wall gap. The results show that as the particle comes close to the wall, its finite size has an increasingly important effect on its own transient motion and the point-dipole approximation may lead to significant error.     

Lithium aminoborohydrides 17. Palladium catalyzed borylation of aryl iodides, bromides, and triflates with diisopropylaminoborane prepared from lithium diisopropylaminoborohydride

The Alcaraz-Vaultier borylation of aryl halides and triflates is reported utilizing diisopropylaminoborane (BH₂N(ⁱPr)₂) prepared from the corresponding lithium aminoborohydride (LAB reagent). BH₂N(ⁱPr)₂, prepared by reacting lithium diisopropylaminoborohydride with trimethylsilyl chloride, provided the most consistent isolated yields from this reaction. Catalytic amounts of palladium dichloride produced the highest yields from aryl iodides, while catalytic tris(dibenzylideneacetone)dipalladium(chloroform) provided the best yields for aryl bromides and triflates. This route to boronic acids is mild enough to tolerate various functionalities and for the first time employs aryl triflates as substrates for the Alcaraz-Vaultier borylation. In addition, it was found that both boronic acid and ester compounds could be isolated from the reaction mixture utilizing simple work-up procedures. Treatment of the reaction intermediate with an acid/base work-up provided the corresponding boronic acid, while treating the same intermediate with a diol, such as neopentyl glycol, afforded the corresponding boronic ester.     

Isolated Cu2+ ions: active sites for selective catalytic reduction of NO

Cu chabazite catalysts show remarkable low temperature activity in selective catalytic reduction (SCR) of NO. This high activity is due to the unique character of the zeolite framework that allows only the presence of one type of isolated mononuclear Cu(2+) species. These Cu(2+) species are the active sites for SCR.     

Isotropic moments over integer lattices

Many modern edge and corner detection algorithms use moment transforms, which convolve images with tensor-valued filters, namely the product of a window function with a monomial. Over continuous domains, one may easily show that such transforms are isotropic. We generalize these continuous results to digital images, that is, to functions over the canonical integer lattice in a finite-dimensional real space. In particular, we first introduce a mathematically well-behaved method for the dilation and rotation of digital images, and then show these operations commute with discrete moment transforms in a manner consistent with the continuous results.     

A library of 3-aryl-4,5-dihydroisoxazole-5-carboxamides

Parallel solution phase methods for the preparation of a 72-membered 3-aryl-4,5-dihydroisoxazole-5-carboxamide library is reported. The reaction order (nitrile oxide 1,3-dipolar cycloaddition followed by amide formation, or vice versa) was investigated both experimentally and computationally to determine which route would result in the highest yields, minimize purification efforts, and give higher 1,3-dipolar cycloaddition regioselectivity. Automated preparative HPLC was used to purify the final products to >or=90% purity on a 10+ mg scale.     

Chemical genomic and proteomic methods for determining kinase inhibitor selectivity

The clinical success of the Bcr-Abl tyrosine kinase inhibitor Gleevec((R)) and the recent clinical approval of a number of small molecule drugs that target protein kinases have intensified the search for novel protein kinase inhibitors. Since most small molecule kinase inhibitors target the highly conserved ATP-binding pocket of this enzyme family, the target selectivity of these molecules is a major concern. Due to the large size of the human kinome, it is a formidable challenge to determine the absolute specificity of a given protein kinase inhibitor, but recent technological developments have made substantial progress in achieving this goal. This review summarizes some of the most recent experimental techniques that have been developed for the determination of protein kinase inhibitor selectivity. Special emphasis is placed on the results of these screens and the general insights that they provide into kinase inhibitor target selectivity.