Critical percolation exploration path and SLE 6: a proof of convergence

It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE ₆. We provide here a detailed proof, which relies on Smirnov’s theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy’s formula). The version of convergence to SLE ₆ that we prove suffices for the Smirnov–Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops. Research of Charles M.Newman was partially supported by the US NSF under grants DMS-01-04278 and DMS-06-06696.     

Diffusion changes in patients with systemic lupus erythematosus

BACKGROUND AND PURPOSE: Systemic lupus erythematosus (SLE) is an autoimmune disease in which almost all the organs are involved. Neuropsychiatric SLE is of one of the major concerns in the clinical evaluation of this disease. Routine magnetic resonance imaging (MRI) findings are often nonspecific or negative. In this study, we explored the use of diffusion tensor imaging in assisting with the diagnosis of SLE. METHODS: Data from 34 SLE patients (age range, 18-73 years) and 29 age-matched volunteers (age range, 29-64 years) were analyzed. MRI was performed on a 1.5-T clinical MR scanner with a quadrature head coil. The average diffusion constant (D(av)) and diffusion anisotropy maps [fractional anisotropy (FA)] were determined on a pixel-by-pixel basis. Regional diffusion measurements were made by region of interest in the genu and splenium of the corpus callosum (CC), anterior and posterior limb of the internal capsule (IC) and frontal lobe and thalamus. The diffusion distribution was fitted to a triple-Gaussian model. The mean of the brain tissue distribution was determined as a mean diffusion constant for the whole brain (BD(av)). Student's t test was used to determine the diffusion difference between SLE patients and control subjects. The SLE patients were separated into two groups according to their MRI results. A P value lower than .05 was considered to be statistically significant. RESULTS: Twenty of the 34 SLE patients with abnormal MRI results showed findings dominated by nonspecific white matter disease. The BD(av) and D(av) values of the frontal lobe, splenium CC and anterior IC were significantly higher in all SLE patients as compared with the control subjects. The SLE patients with normal MRI results also showed higher BD(av) and D(av) values in the frontal lobe, splenium and anterior and posterior limbs of the IC as compared with the control subjects. There was no significant difference in the D(av) values of the thalamus between the SLE patients and the control subjects. The BD(av) value in the SLE patient group was robustly correlated with the D(av) values of the frontal lobe, splenium and thalamus. These correlations were found to be similarly significant for the SLE patients with normal MRI findings. The diffusion anisotropy measurements showed that splenium CC had the highest FA value in both the control subjects and SLE patients. Overall, SLE patients had lower FA values in the genu and splenium CC as compared with the control subjects. In the group of patients with normal MRI findings, the FA values of the genu and splenium CC as well as the anterior IC were also lower than those in the control subjects. Pearson's correlation statistics revealed robust correlations between the measurements of D(av) and FA values in the SLE patient group. CONCLUSION: Quantitative diffusion imaging and diffusion anisotropy showed early changes in the brains of the SLE patients. Increased BD(av) and D(av) values of the frontal lobe as well as decreased anisotropy in the genu CC and anterior IC may represent preclinical signs of central nervous system involvement of SLE even when the routine MRI findings are negative or nonspecific. Quantitative diffusion analysis may prove to be useful in detecting the initial brain involvement of SLE and may enable monitoring of early disease progression and treatment efficacy.     

On radial stochastic Loewner evolution in multiply connected domains

We discuss the extension of radial SLE to multiply connected planar domains. First, we extend Loewner's theory of slit mappings to multiply connected domains by establishing the radial Komatu-Loewner equation, and show that a simple curve from the boundary to the bulk is encoded by a motion on moduli space and a motion on the boundary of the domain. Then, we show that the vector-field describing the motion of the moduli is Lipschitz. We explain why this implies that “consistent,” conformally invariant random simple curves are described by multidimensional diffusions, where one component is a motion on the boundary, and the other component is a motion on moduli space. We argue what the exact form of this diffusion is (up to a single real parameter κ) in order to model boundaries of percolation clusters. Finally, we show that this moduli diffusion leads to random non-self-crossing curves satisfying the locality property if and only if κ=6.     

Solubility of polycyclic aromatics in binary solvent mixtures using activity coefficient models

Polycyclic aromatic hydrocarbons (PAHs) precipitation is one of the major problems in the hydrocracking units. In this investigation, pyrene and phenanthrene were selected because they were found to be in higher concentrations in the feed to hydrocracking units. Their solubilities were investigated in toluene solvent mixture of iso-octane and heptane over a temperature range from 293 to 323 K. The experimental solubility data were used to predict the interaction parameters for seven different solid–liquid equilibrium models. The following activity coefficient models were used; Wilson, NIBS/Redlich–Kister, UNIQUAC, modified UNIFAC, modified UNIFAC (Dortmund), Flory–Huggins and Sheng. The interaction parameters were expressed as a second-order polynomial function in temperature. In order to test the models, the average absolute deviation percentage (AADP) was used. The overall AADP was found to range from approximately 7 to 14%. The models can be arranged according to their accuracy in a descending order based on AADP as follows: NIBS/Redlich–Kister, Wilson, UNIQUAC, Sheng, Flory–Huggins, modified UNIFAC (Dortmund) and finally modified UNIFAC. All models used in this work gave reasonable results; however, the group contribution models can also be used as a predictive tool for the solubility measurement of pyrene and phenanthrene in other solvents containing the same groups of the solvents used in this study.     

Computing the Loewner Driving Process of Random Curves in the Half Plane

We simulate several models of random curves in the half plane and numerically compute the stochastic driving processes that produce the curves through the Loewner equation. Our models include models whose scaling limit is the Schramm-Loewner evolution (SLE) and models for which it is not. We study several tests of whether the driving process is Brownian motion, as it is for SLE. We find that testing only the normality of the process at a fixed time is not effective at determining if the random curves are an SLE. Tests that involve the independence of the increments of Brownian motion are much more effective. We also study the zipper algorithm for numerically computing the driving function of a simple curve. We give an implementation of this algorithm which runs in a time O(N ^1.35) rather than the usual O(N ²), where N is the number of points on the curve.     

Interpreting divergences and Madelung-type sums via extrapolations of sequence generating polynomials: Method of intersection and area rule

Based on explorations in estimating certain Madelung constants, we put forward here two separate strategies to understand the meaning of two distinct classes of divergent non-power-series expansions. One class refers to alternating series representations, the other to monotonic ones. They chiefly rest on precise and approximate polynomial extrapolations, depending on situations. In case of sawtooth sequences, e.g., the partial-sums obtainable from Dirichlet eta or beta function at negative integer arguments, exact sequence-generating polynomials are found. Extrapolations yield a graphical meaning to anti-limit here, along with the exact answer. For staircase sequences, like the ones obtained from partial-sums of series representations for lambda and zeta functions, again at negative integer arguments, anti-limits do not exist. But, correct sequence-generating polynomials are obtained. There, our recipe relies on estimation of specific, finite areas embedded by such polynomials. The schemes put forward here are direct, independent and conceptually appealing. A subsequent extension of the latter strategy to alternating series also lends extra credence. Two new interpretations of summability are gained. Pilot calculations on several types of lattice sums reveal the worth of our endeavor with approximate extrapolations as well.     

A validated solid-liquid extraction method for the HPLC determination of polyphenols in apple tissues Comparison with pressurised liquid extraction

A solid-liquid extraction procedure followed by reversed-phase high-performance liquid chromatography (RP-HPLC) coupled with a photodiode array detector (DAD) for the determination of polyphenols in freeze-dried apple peel and pulp is reported. The extraction step consists in sonicating 0.5 g of freeze-dried apple tissue with 30 mL of methanol-water-acetic acid (30:69:1, v/v/v) containing 2 g of ascorbic acid/L, for 10 min in an ultrasonic bath. The whole method was validated, concluding that it is a robust method that presents high extraction efficiencies (peel: >91%, pulp: >95%) and appropriate precisions (within day: R.S.D. (n = 5) <5%, and between days: R.S.D. (n = 5) <7%) at the different concentration levels of polyphenols that can be found in apple samples. The method was compared with one previously published, consisting in a pressurized liquid extraction (PLE) followed by RP-HPLC-DAD determination. The advantages and disadvantages of both methods are discussed.     

Liquid chromatographic–tandem mass spectrometry assay for the quantification of MDT‐637 in human nasal wash

MDT‐637 is a new chemical entity that shows potent and selective antiviral activity against respiratory syncytial virus. As part of the new drug development program, a method for quantitating MDT‐637 in nasal wash collected from human subjects was required. This article describes the method development and validation and application of a bioanalytical assay for MDT‐637 in human nasal wash using liquid chromatography–tandem mass spectrometry. Sample pretreatment utilized supported liquid extraction and final extracts were injected onto an Agilent Poroshell 120 EC‐C₁₈ column and eluted under isocratic conditions of 65:35 (v/v) 10 mm ammonium formate in water and acetonitrile Detection was performed on an API 5000 tandem mass spectrometer operating in negative electrospray ionization mode applying multiple reaction monitoring. The assay was validated in accordance with the US Food and Drug Administration guidance on bioanalytical method validation and was applied in support of a Phase IIa human safety and tolerability study.     

血浆循环DNA浓度在原发性干燥综合征和系统性红斑狼疮诊治中的意义

目的比较原发干燥综合征（pSS）、系统性红斑狼疮（SLE）患者外周血浆中循环DNA（cDNA）浓度的差异性，探讨cDNA浓度在pSS、SLE诊治中的意义。方法选用人类基因组管家基因甘油醛-3-磷酸脱氢酶（GAPDH）设计引物，测定pSS、SLE及健康对照组中血浆cDNA浓度，同时测定血沉（ESR）、C-反应蛋白（CRP）、免疫球蛋白（IgA、IgG、IgM）并评定不同疾病的活动指数和损害指数，分析血浆cDNA浓度与疾病活动性、相关参数的关系。结果 pSS组cDNA浓度[（368.24±196.68）ng/ml]明显高于SLE组[（244.51±164.10）ng/ml]及健康对照组[（24.56±16.76）ng/ml]，差异均有统计学意义（均P＜0.05）；pSS组中的cDNA浓度值与干燥综合征活动指数（SSDAI）呈正相关（P＜0.05），与干燥综合征损害指数（SSDDI）无相关性（P＞0.05）；SLE组中cDNA浓度值与SLE活动指数（SLEDAI）无相关性（P＞0.05）。结论 cDNA浓度可作为pSS与SLE鉴别指标之一，pSS患者cDNA浓度与SSDAI呈正相关，可作为pSS疾病活动性观察指标之一。