Distributions of Runs and Consecutive Systems on Directed Trees

In this paper we study exact distributions of runs on directed trees. On the assumption that the collection of random variables indexed by the vertices of a directed tree has a directed Markov distribution, the exact distribution theory of runs is extended from based on random sequences to based on directed trees. The distribution of the number of success runs of a specified length on a directed tree along the direction is derived. A consecutive-k-out-of-n:F system on a directed tree is introduced and investigated. By assuming that the lifetimes of the components are independent and identically distributed, we give the exact distribution of the lifetime of the consecutive system. The results are not only theoretical but also suitable for computation.     

Sooner and Later Waiting Time Problems for Runs in Markov Dependent Bivariate Trials

In this paper we study exact distributions of sooner and later waiting times for runs in Markov dependent bivariate trials. We give systems of linear equations with respect to conditional probability generating functions of the waiting times. By considering bivariate trials, we can treat very general and practical waiting time problems for runs of two events which are not necessarily mutually exclusive. Numerical examples are also given in order to illustrate the feasibility of our results.     

Synthesis and properties of new halogen‐substituted polyamides from 5‐halo‐m‐phenylenediamines and both aliphatic and aromatic dicarboxylic acids

New halogen‐substituted aromatic–aliphatic and wholly aromatic polyamides with high inherent viscosities were synthesized by the direct polycondensation of 5‐halo‐m‐phenylenediamines, where the halogens were Cl, Br, and I, with both aliphatic and aromatic dicarboxylic acids in N‐methyl‐2‐pyrrolidone with a mixture of triphenyl phosphite and pyridine as a condensing agent. The solubility of the halogen‐substituted polyamides was much higher than that of the parent polyamides derived from m‐phenylenediamine. The glass‐transition temperatures of the substituted aromatic–aliphatic polyamides increased in the order Cl < Br < I, whereas the temperatures of 10% weight loss in air decreased in the reverse order. The limiting oxygen index values, as an indication of flammability, increased for the substituted aromatic–aliphatic polyamides in the order Cl < Br < I. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 3911–3918, 2000     

Synthesis of hyperbranched poly(ether nitrile)s by one‐step polycondensation of an AB2 monomer

Hyperbranched poly(ether nitrile)s were prepared from a novel AB₂ type monomer, 2‐chloro‐4‐(3,5‐dihydroxyphenoxy)benzonitrile, via nucleophilic aromatic substitution. Soluble and low‐viscous hyperbranched polymers with molecular weights upto 233,600 (M_w) were isolated. According to the ¹H NMR and GPC data, the unique polymerization behavior was observed, which implies that the weight average molecular weight increased after the number average molecular weight reached plateau region. Model compounds were prepared to characterize the branching structure. Spectroscopic measurements of the model compounds and the resulting polymers, such as ¹H, DEPT ¹³C NMR, and MS, strongly suggest that the ether exchange reaction and cyclization are involved in the propagation reaction. The side reactions would affect the unique polymerization behavior. The resulting polymers showed a good solubility in organic solvents similar to other hyperbranched aromatic polymers. The hydroxy‐terminated polymer was even soluble in basic water. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 5835–5844, 2009     

Enhancement of oxygen and methane solubility in 1-hexyl-3-methylimidazolium bis(trifluoromethylsulfonyl) imide using carbon dioxide

The presence of CO(2) increases the solubility of O(2) and CH(4) in 1-hexyl-3-methylimidazolium bis(trifluoromethylsulfonyl) imide at 25 degrees C and pressures to 13 bar.     

Improved method of the MALDI-TOF analysis of DNA with nanodot sample target plate

We report a new sample target plate for MALDI-TOF mass spectrometry made of SiO2 with 30 nm Pt dots for which a highly reproducible and improved DNA analysis was achieved.     

Numbers of Success-Runs of Specified Length Until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k

A new distribution called a generalized binomial distribution of order k is defined and some properties are investigated. A class of enumeration schemes for success-runs of a specified length including non-overlapping and overlapping enumeration schemes is rigorously studied. For each nonnegative integer less than the specified length of the runs, an enumeration scheme called -overlapping way of counting is defined. Let k and be positive integers satisfying < k. Based on independent Bernoulli trials, it is shown that the number of (– 1)-overlapping occurrences of success-run of length k until the n-th overlapping occurrence of success-run of length follows the generalized binomial distribution of order (k–). In particular, the number of non-overlapping occurrences of success-run of length k until the n-th success follows the generalized binomial distribution of order (k– 1). The distribution remains unchanged essentially even if the underlying sequence is changed from the sequence of independent Bernoulli trials to a dependent sequence such as higher order Markov dependent trials. A practical example of the generalized binomial distribution of order k is also given.