A Simple D 2-Sampling Based PTAS for k-Means and Other Clustering Problems

Given a set of points \(P \subset\mathbb{R}^{d}\) , the k-means clustering problem is to find a set of k centers \(C = \{ c_{1},\ldots,c_{k}\}, c_{i} \in\mathbb{R}^{d}\) , such that the objective function ∑ _x∈P e(x,C)², where e(x,C) denotes the Euclidean distance between x and the closest center in C, is minimized. This is one of the most prominent objective functions that has been studied with respect to clustering. D ²-sampling (Arthur and Vassilvitskii, Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’07, pp. 1027–1035, SIAM, Philadelphia, 2007) is a simple non-uniform sampling technique for choosing points from a set of points. It works as follows: given a set of points \(P \subset\mathbb{R}^{d}\) , the first point is chosen uniformly at random from P. Subsequently, a point from P is chosen as the next sample with probability proportional to the square of the distance of this point to the nearest previously sampled point. D ²-sampling has been shown to have nice properties with respect to the k-means clustering problem. Arthur and Vassilvitskii (Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’07, pp. 1027–1035, SIAM, Philadelphia, 2007) show that k points chosen as centers from P using D ²-sampling give an O(logk) approximation in expectation. Ailon et al. (NIPS, pp. 10–18, 2009) and Aggarwal et al. (Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pp. 15–28, Springer, Berlin, 2009) extended results of Arthur and Vassilvitskii (Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’07, pp. 1027–1035, SIAM, Philadelphia, 2007) to show that O(k) points chosen as centers using D ²-sampling give an O(1) approximation to the k-means objective function with high probability. In this paper, we further demonstrate the power of D ²-sampling by giving a simple randomized (1+?)-approximation algorithm that uses the D ²-sampling in its core.     

Towards identifying the roll motion parameters of a motorcycle simulator

This study aimed at identifying the roll motion parameters of a motorcycle simulator prototype. Experienced motorcyclists tuned the angular physical movement of the mock-up and that of the visual scene to achieve an optimal riding experience during curves. The participants exceeded the rolling angles that would be required in real-world riding, while avoiding leaning the mock-up beyond 10°. In addition, they were more influenced by the speed of the virtual motorcycle than by road curvature, especially in a wide field of view. Heterogeneity was found in the roll applied to the visual scene. The overall patterns suggest that at least when washout is not applied to remove the side forces that in real-world riding are compensated by a centrifugal force, greater roll of the visual at the expense of the mock-up is mandatory to avoid performance biases that might be enhanced due to fear of falling off the simulator. Future roll motion models must take into consideration factors such as riding postures, which might not only influence the forces operating on the rider-motorcycle system, but also how motorcyclists perceive the visual world.     

Compositional trend analysis on poly(phenylene ether) based thermoplastic elastomers

Detailed statistical trend analysis of thermoplastic elastomers based on poly (phenylene ether) (PPE), polystyrene (PS), ethylene vinyl acetate (EVA) and styrene‐ethylene‐butylene‐styrene (SEBS) was done through Design Expert software by Stat‐Ease. D optimal crossed design was followed to capture the interaction with the parameters. Effect of blend ratio, vinyl acetate (VA) content of EVA, molecular weight (MW) of SEBS and intrinsic viscosity (IV) of PPE on the blend performance (response) was studied in detail. Design of Experiment (DOE) analysis showed the “optimized formulation” of the blend. Increase in PPE‐polystyrene (PS) content increased tensile strength and modulus of the blend, followed by a decrease in strain at break. However, EVA had a reverse effect on tensile strength and modulus. Strain at break increased significantly with increasing SEBS content in the blend. Graphical and numerical optimization showed that superior mechanical properties (tensile strength, strain at break and modulus) could be achieved at VA content ～ 50% at a particular loading of EVA. Low MW SEBS was found to be more compatible with the other components of the blend. Mechanical properties of the quaternary blend were marginally affected with change in IV of PPE in the range of 0.33 to 0.46. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci 2007     

Synthesis of ethyl propionate catalyzed by poly(N‐AEAAm‐co‐AAc)‐cl‐MBAm hydrogel‐immobilized lipase of Bacillus coagulans MTCC‐6375

A purified alkaline thermotolerant bacterial lipase of Bacillus coagulans MTCC‐6375 was efficiently immobilized onto poly(N‐AEAAm‐co‐AAc‐cl‐MBAm)‐hydrogel at pH 8.5 and at temperature 55°C in 16 h. The hydrogel‐bound matrix possessed 1.04 U/g (matrix) lipase activity with a specific activity of 1.8 U/mg of protein. The immobilized lipase resulted in formation of 52.5 mM of ethyl propionate (52% conversion) at 55°C in 9 h in n‐nonane. Ethanol and propionic acid when used in a ratio of 300 : 100 mM, respectively, in n‐nonane along with 10 mg of hydrogel‐bound lipase resulted in optimal synthesis of ethyl propionate (82.5 mM). Addition of molecular sieves (3 Å, 0.7 g/reaction volume) further enhanced the conversion rate to 82.4% resulting in 83.5 mM of ethyl propionate. Incubation temperature below or above 55°C had a marked effect on the synthesis of ethyl propionate. However, esterification performed in n‐heptane at 65°C resulted in 87.5 mM of ethyl propionate with a conversation rate of 89.3%. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci 2007     

Prediction of the hydraulic diffusivity from pore size distribution of concrete

A model is presented in this work through which variation of hydraulic diffusivity of concrete with relative water content can be obtained from pore size distribution as an input. The specific water capacity and hydraulic conductivity of concrete are expressed in terms of pore size characteristics, considering laminar flow due to capillary suction through tortuous elliptic tubes, oriented equally in three orthogonal directions. Hydraulic diffusivity being the ratio of hydraulic conductivity and specific water capacity is thus expressed in terms of pore size characteristics. The input pore size distributions have been determined experimentally for normal strength concrete mixes through mercury intrusion porosimetry. Using the model the variation of hydraulic diffusivity with relative water content is determined for three cases viz. 1) ideal continuous wetting, 2) ideal continuous drying and 3) random access of pores by water. These results are then compared with an experimentally obtained variation.     

Preparation and Crystal Structure of KFeCl3 and KFeBr3

The preparation and crystal structures of anhydrous KFeCl₃ and KFeBr₃ are described. X-ray diffraction data obtained at room temperature indicate that the two compounds belong to the orthorhombic system, space group Pnma and z = 4. Unit cell parameters are a = 8.712 Å, b = 3.845 Å, c = 14.15 Å for KFeCl₃ and a = 9.220 Å, b = 4.026 Å, c = 14.899 Å for KFeBr₃. Atomic positions were determined.     

Effect of Precursor and Solvent on Morphology of Zirconia Nanoparticles Produced by Combustion Aerosol Synthesis

Zirconia nanoparticles were synthesized using a flame-based system involving spray droplet combustion of different precursor solutions. The characteristics of the feed were varied by varying the precursor compound, precursor concentration, and solvent type, and by using droplets of different mean sizes. When large droplets were used, agglomerated particles were formed when an organometallic precursor was used and large cenospheric particles were produced when an inorganic precursor was used. Reduction of the droplet size to a number-mean droplet diameter of 3.2 μm resulted in the production of solid spherical particles regardless of the precursor type. When an inertial impactor was used to eliminate droplets larger than 2.3 μm, the large particles in the final product were eliminated and uniformly sized solid zirconia particles having a smaller mean size were produced. The final particle size did not vary with the concentration of the precursor, indicating that multiple ceramic particles resulted from each precursor-containing droplet.     

Prediction of Experimental Methanol Decomposition Rates on Platinum from First Principles

A microkinetic model for methanol decomposition on platinum is presented. The model incorporates competitive decomposition pathways, beginning with both O–H and C–H bond scission in methanol, and uses results from density functional theory (DFT) calculations [Greeley and Mavrikakis, J. Am. Chem. Soc. 124 (2002) 7193, Greeley and Mavrikakis, J. Am. Chem. Soc. 126 (2004) 3910]. Results from reaction kinetics experiments show that the rate of H₂ production increases with increasing temperature and methanol concentration in the feed and is only nominally affected by the presence of CO or H₂ with methanol. The model, based on the values of binding energies, pre-exponential factors and activation energy barriers derived from first principles calculations, accurately predicts experimental reaction rates and orders. The model also gives insight into the most favorable reaction pathway, the rate-limiting step, the apparent activation energy, coverages, and the effects of pressure. It is found that the pathway beginning with the C–H bond scission (CH₃OH→H₂COH→HCOH→CO) is dominant compared with the path beginning with O–H bond scission. The cleavage of the first C–H bond in methanol is the rate-controlling step. The surface is highly poisoned by CO, whereas COH appears to be a spectator species.     

Initial Deposition of Colloidal Particles on a Rough Nanofiltration Membrane

The initial rate of colloid deposition onto semi‐permeable membranes is largely controlled by the coupled influence of permeation drag and particle‐membrane colloidal interactions. Recent studies show that the particle‐membrane interactions are subject to immense local variations due to the inherent morphological heterogeneity (roughness) of reverse osmosis (RO) and nanofiltration (NF) membranes. This experimental investigation reports the effect of membrane roughness on the initial deposition of polystyrene latex particles on a rough NF membrane during cross flow membrane filtration under different operating pressures and solution chemistries. Atomic force microscopy was used to characterize the roughness of the membrane and observe the structure of particle deposits. At the initial stages of fouling, the AFM images show that more particles preferentially accumulate near the “peaks” than in the “valleys” of the rough NF membrane surface.     

Melt‐processable rubber: Chlorinated waste tire rubber‐filled polyvinyl chloride

Chlorinated ground rubber tire (Cl‐GRT) particles were used as filler in a plasticized polyvinylchloride (PVC) to develop a melt‐processable rubber composition. Physical properties of the Cl‐GRT‐filled PVC compound showed improvement compared to the nonchlorinated counterpart. Interaction between Cl‐GRT and PVC was examined on the basis of results of stress relaxation, dynamic mechanical thermal analysis, and solvent swelling studies. The Cl‐GRT could be loaded upto 40 parts per hundred parts of PVC, and the composition still retains the elastomeric characteristics. The Cl‐GRT‐filled composite was found to be reprocessable like the unfilled PVC compound. © 2002 Wiley Periodicals, Inc. J Appl Polym Sci 84: 622–631, 2002; DOI 10.1002/app.10352