Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q|=1 and Quantum Dilogarithm

Several kinds of q-orthogonal polynomials with |q|=1 are constructed as the main parts of the eigenfunctions of new solvable discrete quantum mechanical systems. Their orthogonality weight functions consist of quantum dilogarithm functions, which are a natural extension of the Euler gamma functions and the q-gamma functions (q-shifted factorials). The dimensions of the orthogonal spaces are finite. These q-orthogonal polynomials are expressed in terms of the Askey-Wilson polynomials and their certain limit forms.Comment: 37 pages. Comments and references added. To appear in J.Math.Phy

Tracking ‘learning behaviours’ in the incidental acquisition of two dimensional adjectives by Japanese beginner learners of L2 English

The purpose of this article is to examine both the process and product of vocabulary learning in a task-based instructional context. The article reports a study that investigated the acquisition of two dimensional adjectives (‘big’ and ‘small’) by six-year-old Japanese children who were complete beginners. It tracked the ‘learning behaviours’ that occurred in the classroom interactions involving the use of these adjectives in nine task-based lessons to show how these behaviours developed over time. It also collected test data to establish whether the learners had developed the receptive and productive knowledge required for the independent use of two adjectives. In this way, the study shows how second language (L2) learning evolves through interaction by exploring the relationships between the learners’ different learning behaviours and the differences in their test performance. The main finding was that differences in the success of the individual learners in acquiring productive control over the dimensional adjectives – as shown in the tests – was directly traceable to their learning behaviours in the task-based interactions

The comparative effect of direct written corrective feedback and metalinguistic explanation on learners' explicit and implicit knowledge of the English indefinite article

The study extends current work on written error feedback in writing in two ways. First, it examines whether it has an effect on adult ESL learners' L2 implicit and explicit knowledge. Second, the study compares the effect of one common type of feedback - direct corrective feedback (DCF) - with an alternative type of error feedback - the provision of metalinguistic explanation (ME). The effect of these two types of error feedback was measured by an Error Correction Test (ECT) and by examining the accuracy of use of the target feature (the English indefinite article) in both a revised text and in new pieces of writing by 49 low-intermediate ESL students in an intensive language programme in the United States. In addition, eye-tracking data and self-reports elicited from the learners provided information about the use that they made of the DCF and ME. It was found that the DCF had no effect on accurate use of the target feature suggesting that it benefited neither implicit nor explicit knowledge. In contrast, the ME led to gains in accuracy in the ECT and in a new piece of writing completed immediately after the treatment but not in a second new text completed two weeks later. These results are interpreted as indicating that the ME helped to develop learners' L2 explicit knowledge but that the effect was not durable and thus probably had no effect on their implicit knowledge. Learners' self-reports indicate that the learners receiving the DCF did not develop awareness of the rule whereas those receiving the ME did and were able to use it when revising their original text. These findings are discussed from the perspective of both SLA theory and language pedagogy and suggestions for further research are put forward

Neutron electric dipole moment with external electric field method in lattice QCD

We discuss a possibility that the Neutron Electric Dipole Moment (NEDM) can be calculated in lattice QCD simulations in the presence of the CP violating $\theta$ term. In this paper we measure the energy difference between spin-up and spin-down states of the neutron in the presence of an uniform and static external electric field. We first test this method in quenched QCD with the RG improved gauge action on a $16^3\times 32$ lattice at $a^{-1}\simeq$ 2 GeV, employing two different lattice fermion formulations, the domain-wall fermion and the clover fermion for quarks, at relatively heavy quark mass $(m_{PS}/m_V \simeq 0.85)$. We obtain non-zero values of NEDM from calculations with both fermion formulations. We next consider some systematic uncertainties of our method for NEDM, using $24^3\times 32$ lattice at the same lattice spacing only with the clover fermion. We finally investigate the quark mass dependence of NEDM and observe a non-vanishing behavior of NEDM toward the chiral limit. We interpret this behavior as a manifestation of the pathology in the quenched approximation.Comment: LaTeX2e, 51 pages, 43 figures, uses revtex4 and graphicx, References and comments added, typos corrected, accepted by PR