Crossing numbers of sequences of graphs II: Planar tiles

We describe a method of creating an infinite family of crossing‐critical graphs from a single small planar map, the tile, by gluing together many copies of the tile together in a circular fashion. This method yields all known infinite families of k‐crossing‐critical graphs. Furthermore, the method yields new infinite families, which extend from (4,6) to (3.5,6) the interval of rationals r for which there is, for some k, an infinite sequence of k‐crossing‐critical graphs all having average degree r. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 332–341, 2003     

Gas permeability and n‐butane solubility of poly(1‐trimethylgermyl‐1‐propyne)

The gas permeability and n‐butane solubility in glassy poly(1‐trimethylgermyl‐1‐propyne) (PTMGP) are reported. As synthesized, the PTMGP product contains two fractions: (1) one that is insoluble in toluene and soluble only in carbon disulfide (the toluene‐insoluble polymer) and (2) one that is soluble in both toluene and carbon disulfide (the toluene‐soluble polymer). In as‐cast films, the gas permeability and n‐butane solubility are higher in films prepared from the toluene‐soluble polymer (particularly in those films cast from toluene) than in films prepared from the toluene‐insoluble polymer and increase to a maximum in both fractions after methanol conditioning. For example, in as‐cast films prepared from carbon disulfide, the oxygen permeability at 35 °C is 330 × 10^?10 cm³ (STP) cm/(cm² s cmHg) for the toluene‐soluble polymer and 73 × 10^?10 cm³ (STP) cm/(cm² s cmHg) for the toluene‐insoluble polymer. After these films are conditioned in methanol, the oxygen permeability increases to 5200 × 10^?10 cm³ (STP) cm/(cm² s cmHg) for the toluene‐soluble polymer and 6200 × 10^?10 cm³ (STP) cm/(cm² s cmHg) for the toluene‐insoluble polymer. The rankings of the fractional free volume and nonequilibrium excess free volume in the various PTMGP films are consistent with the measured gas permeability and n‐butane solubility values. Methanol conditioning increases gas permeability and n‐butane solubility of as‐cast PTMGP films, regardless of the polymer fraction type and casting solvent used, and minimizes the permeability and solubility differences between the various films (i.e., the permeability and solubility values of all conditioned PTMGP films are similar). © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 2228–2236, 2002     

Evaluation of extraction methods for quantification of aqueous fullerenes in urine

There is a growing concern about the human and environmental health effects of fullerenes (e.g., C(60)) due to their increasing application in research, medicine, and industry. Toxicological and pharmacokinetic research requires standard methods for extraction and detection of fullerenes from biological matrices such as urine. The present study validates the use of liquid-liquid extraction (LLE) and solid-phase extraction (SPE) methods in conjunction with liquid chromatography-mass spectrometry (LC-MS) for the quantitative determination of C(60) in human and synthetic urine as compared with ultrapure water. Glacial acetic acid, which is necessary to prevent emulsions during LLE, inhibited C(60) detection by LC-MS, but this could be mitigated with evaporation. Aqueous C(60) aggregates (nC(60)) were spiked at 180 μg/L into the components of a synthetic urine recipe to determine their individual impacts on extraction and detection. Urea, creatinine, and a complex protein (i.e., gelatin) were found to impair SPE, leading to a low recovery rate of 43 ± 4% for C(60) spiked into human urine. In contrast, C(60) was consistently recovered from synthetic matrices using LLE, and recovery in human urine was 80 ± 6%. These results suggest that LLE combined with LC-MS is suitable for studying the clearance of fullerenes from the body. LLE is a robust technique that holds promise for extracting C(60) from other complex biological matrices (e.g., blood, sweat, amniotic fluid) in toxicological studies, enabling a better understanding of the behavior of fullerenes in human and animal systems and facilitating a more comprehensive risk evaluation of fullerenes.     

A conjecture of Erd?s on graph Ramsey numbers

The Ramsey number r(G) of a graph G is the minimum N such that every red–blue coloring of the edges of the complete graph on N vertices contains a monochromatic copy of G. Determining or estimating these numbers is one of the central problems in combinatorics.One of the oldest results in Ramsey Theory, proved by Erd?s and Szekeres in 1935, asserts that the Ramsey number of the complete graph with m edges is at most . Motivated by this estimate Erd?s conjectured, more than a quarter century ago, that there is an absolute constant c such that for any graph G with m edges and no isolated vertices. In this short note we prove this conjecture.     

A New Lower Bound For A Ramsey-Type Problem

Let 3 ≤ r < s be fixed integers and let G be a graph on n vertices not containing a complete graph on s vertices. The main aim of this paper is to provide a new lower bound on the size of the maximum subset of G without a copy of complete graph K_r. Our results substantially improve previous bounds of Krivelevich and Bollobás and Hind. * Research supported in part by NSF grants DMS-0106589, CCR-9987845 and by the State of New Jersey. Part of this research was done while visiting Microsoft Research.     

Prediction of a linear polymer made of xenon and carbon

Electronic structure calculations predict the existence of a novel type of a chemically bound noble gas compound. The predicted species is an extended linear and periodic polymer, made of the repeat unit -(XeCC)-, where CC is the acetylenic group. The polymer has a strong partly ionic nature, with positive partial charge on the xenon atoms and a negative one on the CC groups. High energy barriers are found for the removal of a Xe atom from the chain, indicating high stability. This is the first polymer with a noble-gas-containing building block.     

Cr(VI) adsorption mechanism on rice husk ash burned at low temperature by method of IR spectra

Boehm titration method was used to analyze functional groups on cell surface of rice husk ash burned at low temperature in the present paper. Effects of initial pH value and temperature on Cr(VI) adsorption were studied, adsorption capacity was tested with the help of kinetic models and adsorption isotherms, instruments of Fourier transform infrared spectroscopy(FTIR) and scanning electron microscope (SEM) were used to check characteristics and adsorption mechanism of Cr(VI). The results indicated that optimal removing rate was achieved at initial pH value 5, and pH values of aqueous solution changed little be fore and after adsorption process. The adsorbent of rice husk ash could remove Cr(VI) effectively, and the maximum removing rate could be 95% with Cr(VI) concentration 20 mg x L(-1) and achieve 1-2 level of state standard(GB8978-1996). The adsorption process fits pseudo-second-order kinetic model and Langmuir isotherm better, the maximum adsorption capacity of Cr(VI) was 3.2776 mg x g(-1). Results of FTIR showed that amide II band, Si--O--Si, O--Si--O were important for Cr(VI) removal. SEM micrographs revealed that series of needle-shaped precipitation appeared on cell surface, and inorganic precipitation mechanism and redox mechanism might work in the test. As a kind of low cost adsorbent, rice husk ash can be applied to remove heavy metals in environment with great potential.     

Asymptotics in percolation on high‐girth expanders

We consider supercritical bond percolation on a family of high‐girth ‐regular expanders. The previous study of Alon, Benjamini and Stacey established that its critical probability for the appearance of a linear‐sized (“giant”) component is . Our main result recovers the sharp asymptotics of the size and degree distribution of the vertices in the giant and its 2‐core at any . It was further shown in the previous study that the second largest component, at any , has size at most for some . We show that, unlike the situation in the classical Erd?s‐Rényi random graph, the second largest component in bond percolation on a regular expander, even with an arbitrarily large girth, can have size for arbitrarily close to 1. Moreover, as a by‐product of that construction, we answer negatively a question of Benjamini on the relation between the diameter of a component in percolation on expanders and the existence of a giant component. Finally, we establish other typical features of the giant component, for example, the existence of a linear path.     

Local resilience of graphs

In this article, we initiate a systematic study of graph resilience. The (local) resilience of a graph G with respect to a property measures how much one has to change G (locally) to destroy . Estimating the resilience leads to many new and challenging problems. Here we focus on random and pseudorandom graphs and prove several sharp results. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

On the number of significant mode-mode anharmonic couplings in vibrational calculations: Correlation-corrected vibrational self-consistent field treatment of di-, tri-, and tetrapeptides

A computational study is made of the number of important anharmonic mode-mode couplings in the context of vibrational calculations for di-, tri-, and tetrapeptides. The method employed is the correlation-corrected vibrational self-consistent field (CC-VSCF) algorithm, which includes correlation effects between different vibrational modes. It is found that results of good accuracy can be obtained in calculations that include only N log N mode-mode coupling terms, where N is the number of modes. This simplification significantly accelerates CC-VSCF calculations for large molecules. A criterion based on the characteristics of the normal-mode displacements is employed to predict a priori unimportant coupling terms. The criterion is tested statistically using Spearman's rank correlation coefficient. The results are illustrated by calculations for several di-, tri-, and tetrapeptides using semiempirical PM3 potential surfaces. These results are analyzed and a statistical model for error estimation is given. The decrease in the number of included coupling from N(2) to N log N opens possibilities of anharmonic vibrational calculations for large peptides.