Asymptotic Gap Probability Distributions of the Gaussian Unitary Ensembles and Jacobi Unitary Ensembles

In this paper, we address a class of problems in unitary ensembles. Specifically, we study the probability that a gap symmetric about 0, i.e. $(-a,a)$ is found in the Gaussian unitary ensembles (GUE) and the Jacobi unitary ensembles (JUE) (where in the JUE, we take the parameters $\alpha=\beta$). By exploiting the even parity of the weight, a doubling of the interval to $(a^2,\infty)$ for the GUE, and $(a^2,1)$, for the (symmetric) JUE, shows that the gap probabilities maybe determined as the product of the smallest eigenvalue distributions of the LUE with parameter $\alpha=-1/2,$ and $\alpha=1/2$ and the (shifted) JUE with weights $x^{1/2}(1-x)^{\beta}$ and $x^{-1/2}(1-x)^{\beta}$ The $\sigma$ function, namely, the derivative of the log of the smallest eigenvalue distributions of the finite-$n$ LUE or the JUE, satisfies the Jimbo-Miwa-Okamoto $\sigma$ form of $P_{V}$ and $P_{VI}$, although in the shift Jacobi case, with the weight $x^{\alpha}(1-x)^{\beta},$ the $\beta$ parameter does not show up in the equation. We also obtain the asymptotic expansions for the smallest eigenvalue distributions of the Laguerre unitary and Jacobi unitary ensembles after appropriate double scalings, and obtained the constants in the asymptotic expansion of the gap probablities, expressed in term of the Barnes $G-$ function valuated at special point.Comment: 38 page

Painlev\'e III′' and the Hankel Determinant Generated by a Singularly Perturbed Gaussian Weight

In this paper, we study the Hankel determinant generated by a singularly perturbed Gaussian weight $$w(x,t)=\mathrm{e}^{-x^{2}-\frac{t}{x^{2}}},\;\;x\in(-\infty, \infty),\;\;t>0.$$ By using the ladder operator approach associated with the orthogonal polynomials, we show that the logarithmic derivative of the Hankel determinant satisfies both a non-linear second order difference equation and a non-linear second order differential equation. The Hankel determinant also admits an integral representation involving a Painlev\'e III$'$. Furthermore, we consider the asymptotics of the Hankel determinant under a double scaling, i.e. $n\rightarrow\infty$ and $t\rightarrow 0$ such that $s=(2n+1)t$ is fixed. The asymptotic expansions of the scaled Hankel determinant for large $s$ and small $s$ are established, from which Dyson's constant appears.Comment: 22 page

China's Innovation System and the Move Toward Harmonious Growth and Endogenous Innovation

Observers around the world are impressed by the rapid growth of China’s economy. While outside observers tend to focus on the success story of unprecedented growth policy documents and recent domestic debates in China have pointed to the need for a shift in the growth trajectory with stronger emphasis on ‘endogenous innovation’ and ‘harmonious development’. This paper attempts to capture the current characteristics of China’s production and innovation system; how they were shaped by history and what major challenges they raise for the future. On the basis of the analysis the authors propose that it is possible to link together the two key concepts ‘endogenous innovation’ and ‘harmonious development’ by focusing innovation and development efforts in China on domestic needs, including social needs, rather than a one-sided focus on export-promotion and commodity production.China; economic growth; R&D; innovation systems

Estudio exploratorio sobre el papel que desempeña la utilización de la L1 en los comentarios escritos por compañeros acerca de la escritura de la L2

While research on peer feedback in L2 writing has proliferated in the past two decades, little research has investigated the role of language use in peer written feedback in EFL writing contexts. The existing peer feedback studies have failed to explain how the use of L1 and L2 in peer written feedback can influence the peer feedback practices. Using the peer written feedback on a survey report by Chinese EFL learners as the data, the current study aims to investigate how the use of L1 in the process of giving feedback influences the peer written feedback practices. The findings indicate that L1 could play a mediating role in the written peer feedback process and L1 was found to help mediate the cognitive resources in working memory and bring positive changes to the feedback practices. With the scaffolding role of L1, L2 learners could produce more comments on content and offer more specific feedback. Implications and suggestions regarding the use of L1 in peer feedback activity have been proposed finally.Si bien estudios anteriores han contribuido a que comprendamos la naturaleza de los comentarios que realizan otros compañeros, los efectos mediadores del idioma en los procesos de retroalimentación entre compañeros para alumnos de ESL/EFL no se han investigado lo suficiente. Los estudios actuales en esta materia no han conseguido explicar de qué modo pueden influir la utilización de la L1 y L2 en la comunicación oral y los comentarios escritos que realizan otros compañeros en las prácticas de retroalimentación. Tomando como base los comentarios escritos por estudiantes chinos de EFL en una encuesta, este estudio tiene como objetivo investigar cómo influye el uso de la L1 en los procesos de retroalimentación en el modo en que los compañeros redactan sus observaciones. Los resultados muestran que la L1 podría desempeñar un papel mediador en los procesos de retroalimentación escrita de compañeros, además de que puede ayudar a mediar en los recursos cognitivos en la memoria de trabajo y aportar cambios positivos a las prácticas de retroalimentación. Gracias a la función de apoyo que ofrece la L1, los estudiantes de la L2 podrían hacer más observaciones sobre el contenido y proporcionar comentarios más específicos. Por fin se han tenido en cuenta las implicaciones y sugerencias en cuanto al uso de la L1 en la redacción de comentarios por parte de compañeros