Independence ratio and random eigenvectors in transitive graphs

A theorem of Hoffman gives an upper bound on the independence ratio of regular graphs in terms of the minimum $\lambda_{\min}$ of the spectrum of the adjacency matrix. To complement this result we use random eigenvectors to gain lower bounds in the vertex-transitive case. For example, we prove that the independence ratio of a $3$-regular transitive graph is at least $$q=\frac{1}{2}-\frac{3}{4\pi}\arccos\biggl(\frac{1-\lambda _{\min}}{4}\biggr).$$ The same bound holds for infinite transitive graphs: we construct factor of i.i.d. independent sets for which the probability that any given vertex is in the set is at least $q-o(1)$. We also show that the set of the distributions of factor of i.i.d. processes is not closed w.r.t. the weak topology provided that the spectrum of the graph is uncountable.Comment: Published at http://dx.doi.org/10.1214/14-AOP952 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Invariant Gaussian processes and independent sets on regular graphs of large Girth

We prove that every 3-regular, n-vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.) In fact, computer simulation suggests that the bound our method provides is about 0.438n. Our method uses invariant Gaussian processes on the d-regular tree that satisfy the eigenvector equation at each vertex for a certain eigenvalue λ. We show that such processes can be approximated by i.i.d. factors provided that |λ|≤2d-1. We then use these approximations for λ=-2d-1 to produce factor of i.i.d. independent sets on regular trees. © 2014 Wiley Periodicals, Inc

Entropy and expansion

Shearer's inequality bounds the sum of joint entropies of random variables in terms of the total joint entropy. We give another lower bound for the same sum in terms of the individual entropies when the variables are functions of independent random seeds. The inequality involves a constant characterizing the expansion properties of the system. Our results generalize to entropy inequalities used in recent work in invariant settings, including the edge-vertex inequality for factor-of-IID processes, Bowen's entropy inequalities, and Bollob\'as's entropy bounds in random regular graphs. The proof method yields inequalities for other measures of randomness, including covariance. As an application, we give upper bounds for independent sets in both finite and infinite graphs

C‐3 epimers of sugar amino acids as foldameric building blocks: improved synthesis, useful derivatives, coupling strategies

To obtain key sugar derivatives for making homooligomeric foldamers or α/β-chimera peptides, economic and multigram scale synthetic methods were to be developed. Though described in the literature, the costeffective making of both 3-amino-3-deoxy-ribofuranuronic acid (H–tX–OH) and its C-3 epimeric stereoisomer, the 3-amino-3-deoxy-xylofuranuronic acid (H–cX–OH) from d-glucose is described here. The present synthetic route elaborated is (1) appropriate for large-scale synthesis; (2) reagent costs reduced (e.g. by a factor of 400); (3) yields optimized are ~80% or higher for all six consecutive steps concluding –tX– or –cX– and (4) reaction times shortened. Thus, a new synthetic route step-by-step optimized for yield, cost, time and purification is given both for d-xylo and d-ribo-amino-furanuronic acids using sustainable chemistry (e.g. less chromatography with organic solvents; using continuous-flow reactor). Our study encompasses necessary building blocks (e.g. –X–OMe, –X–OiPr, –X–NHMe, Fmoc–X–OH) and key coupling reactions making –Aaa–tX–Aaa– or –Aaa–tX–tX–Aaa– type “inserts”. Completed for both stereoisomers of X, including the newly synthesized Fmoc–cX–OH, producing longer oligomers for drug design and discovery is more of a reality than a wish

Origin of problems related to Staudinger reduction in carbopeptoid syntheses

Abstract We report the solid phase synthesis of –GG-X-GG– type α/β-carbopeptoids incorporating RibAFU(ip) (1a, tX) or XylAFU(ip) (2a, cX) sugar amino acids. Though coupling efficacy is moderate, both the lengthier synthetic route using Fmoc derivative (e.g., Fmoc-RibAFU(ip)-OH) and the azido derivative (e.g., N3-RibAFU(ip)-OH) via Staudinger reaction with nBu3P can be successfully applied. Both X-ray diffraction, 1H- and 31P-NMR, and theoretical (QM) data support and explain why the application of Ph3P as Staudinger reagent is “ineffective” in the case of a cis stereoisomer, if cX is attached to the preceding residue with a peptide (–CONH–) bond. The failure of the polypeptide chain elongation with N3-cX originates from the “coincidence” of a steric crowdedness and an electronic effect disabling the mandatory nucleophilic attack during the hydrolysis of a quasi penta-coordinated triphenylphosphinimine. Nevertheless, the synthesis of the above α/β-chimera peptides as completed now by a new pathway via 1,2-O-isopropylidene-3-azido-3-deoxy-ribo- and -xylo-furanuronic acid (H-RibAFU(ip)-OH 1a and H-XylAFU(ip)-OH 2a) coupled with N-protected α-amino acids on solid phase could serve as useful examples and starting points of further synthetic efforts

Diclofenac Prolongs Repolarization in Ventricular Muscle with Impaired Repolarization Reserve

Background: The aim of the present work was to characterize the electrophysiological effects of the non-steroidal anti- inflammatory drug diclofenac and to study the possible proarrhythmic potency of the drug in ventricular muscle. Methods: Ion currents were recorded using voltage clamp technique in canine single ventricular cells and action potentials were obtained from canine ventricular preparations using microelectrodes. The proarrhythmic potency of the drug was investigated in an anaesthetized rabbit proarrhythmia model. Results: Action potentials were slightly lengthened in ventricular muscle but were shortened in Purkinje fibers by diclofenac (20 mM). The maximum upstroke velocity was decreased in both preparations. Larger repolarization prolongation was observed when repolarization reserve was impaired by previous BaCl 2 application. Diclofenac (3 mg/kg) did not prolong while dofetilide (25 mg/kg) significantly lengthened the QT c interval in anaesthetized rabbits. The addition of diclofenac following reduction of repolarization reserve by dofetilide further prolonged QT c . Diclofenac alone did not induce Torsades de Pointes ventricular tachycardia (TdP) while TdP incidence following dofetilide was 20%. However, the combination of diclofenac and dofetilide significantly increased TdP incidence (62%). In single ventricular cells diclofenac (30 mM) decreased the amplitude of rapid (I Kr ) and slow (I Ks ) delayed rectifier currents thereby attenuating repolarization reserve. L-type calcium current (I Ca ) was slightly diminished, but the transient outward (I to ) and inward rectifier (I K1 ) potassium currents were not influenced. Conclusions: Diclofenac at therapeutic concentrations and even at high dose does not prolong repolarization markedly and does not increase the risk of arrhythmia in normal heart. However, high dose diclofenac treatment may lengthen repolarization and enhance proarrhythmic risk in hearts with reduced repolarization reserve

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were found to be associated with infection-induced asthma [p = 0.0009, OR (95% CI) = 0.5 (0.4-0.8)]. The rs487202-rs574913 CA haplotype was more frequent among patients with infection-induced asthma [p = 0.0006, OR (95% CI) = 1.9 (1.3-2.6)]. None of the SNPs contributed directly to the risk of asthma. Conclusions: Our results suggest that genetic variation in the HRH4 gene might influence the pathogenesis of infection-induced asthma