On-lattice coalescence and annihilation of immobile reactants in loopless lattices and beyond.
We study the behavior of the chemical reactions A+A-->A+S and A+A-->S+S (where the reactive species A and the inert species S are both assumed to be immobile) embedded on Bethe lattices of arbitrary coordination number z and on a two-dimensional (2D) square lattice. For the Bethe lattice case, exact solutions for the coverage in the A species in terms of the initial condition are obtained. In particular, our results hold for the important case of an infinite one-dimensional (1D) lattice (z=2). The method is based on an expansion in terms of conditional probabilities which exploits a Markovian property of these systems. Along the same lines, an approximate solution for the case of a 2D square lattice is developed. The effect of dilution in a random initial condition is discussed in detail, both for the lattice coverage and for the spatial distribution of reactants.
Reaction Kinetics of a Long‐Chain Molecule. II. Arends' Solution
In Part I [J. Chem. Phys. 37, 2584 (1962)] we considered the reaction kinetics of a long‐chain molecule, each segment of which carries a reactive group. The reactivity of each group depended upon whether zero, one, or two of its nearest neighbors had reacted. The average fractions of reacted groups and of unreacted with zero, one of two reacted neighbors were found as functions of time. Now we derive C. B. Arends' [J. Chem. Phys. 38, 322 (1963)] solution for the average fraction of sequences of unreacted groups of n for n=1, 2, ···. We start with the infinite set of differential equations for these average fractions derived by W. G. Lloyd and T. Alfrey, Jr. [J. Chem. Phys. 38, 318 (1963)]. We solve them by assuming a solution of the form derived by Arends. Thus we verify that Arends' solution satisfies the equations of Lloyd and Alfrey and we derive it more simply. In addition we show that the present results yield exactly those of Part I, thus verifying the hypothesis introduced in that work.
Mining Words in the Minds of Second Language Learners: Learner-Specific Word Difficulty
While there have been many studies on measuring the size of learners’ vocabulary or the vocabulary they should learn, there have been few studies on what kind of words learners actually know. Therefore, we investigated theoretically and practically important models for predicting second language learners’ vocabulary and propose another model for this vocabulary prediction task. With the current models, the same word difficulty measure is shared by all learners. This is unrealistic because some learners have special interests. A learner interested in music may know special music-related terms regardless of their difficulty. To solve this problem, our model can define a learner-specific word difficulty measure. Our model is also an extension of these current models in the sense that these models are special cases of our model. In a qualitative evaluation, we defined a measure for how learner-specific a word is. Interestingly, the word with the highest learner-specificity was “twitter”. Although “twitter” is a difficult English word, some low-ability learners presumably knew this word through the famous micro-blogging service. Our qualitative evaluation successfully extracted such interesting and suggestive examples. Our model achieved an accuracy competitive with the current models.