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

Gaussian Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlev\'{e} IV System

We study the Hankel determinant generated by the Gaussian weight with jump discontinuities at $t_1,\cdots,t_m$. By making use of a pair of ladder operators satisfied by the associated monic orthogonal polynomials and three supplementary conditions, we show that the logarithmic derivative of the Hankel determinant satisfies a second order partial differential equation which is reduced to the $\sigma$-form of a Painlev\'{e} IV equation when $m=1$. Moreover, under the assumption that $t_k-t_1$ is fixed for $k=2,\cdots,m$, by considering the Riemann-Hilbert problem for the orthogonal polynomials, we construct direct relationships between the auxiliary quantities introduced in the ladder operators and solutions of a coupled Painlev\'{e} IV system

A Novel Method of Failure Sample Selection for Electrical Systems Using Ant Colony Optimization

The influence of failure propagation is ignored in failure sample selection based on traditional testability demonstration experiment method. Traditional failure sample selection generally causes the omission of some failures during the selection and this phenomenon could lead to some fearful risks of usage because these failures will lead to serious propagation failures. This paper proposes a new failure sample selection method to solve the problem. First, the method uses a directed graph and ant colony optimization (ACO) to obtain a subsequent failure propagation set (SFPS) based on failure propagation model and then we propose a new failure sample selection method on the basis of the number of SFPS. Compared with traditional sampling plan, this method is able to improve the coverage of testing failure samples, increase the capacity of diagnosis, and decrease the risk of using

NF-κB mediates the transcription of mouse calsarcin-1 gene, but not calsarcin-2, in C2C12 cells

BACKGROUND: The calsarcins comprise a novel family of muscle-specific calcineurin-interaction proteins that play an important role in modulating both the function and substrate specificity of calcineurin in muscle cells. The expression of calsarcin-1 (CS-1) is restricted to slow-twitch skeletal muscle fibres, whereas that of both calsarcin-2 (CS-2) and calsarcin-3 (CS-3) is enriched in fast-twitch fibres. However, the transcriptional control of this selective expression has not been previously elucidated. RESULTS: Our real-time RT-PCR analyses suggest that the expression of CS-1 and CS-2 is increased during the myogenic differentiation of mouse C2C12 cells. Promoter deletion analysis further suggests that an NF-κB binding site within the CS-1 promoter is responsible for the up-regulation of CS-1 transcription, but no similar mechanism was evident for CS-2. These findings are further supported by the results of EMSA analysis, as well as by overexpression and inhibition experiments in which NF-κB function was blocked by treatment with its inhibitor, PDTC. In addition, the overexpression of NFATc4 induces both the CS-1 and CS-2 promoters, whereas MEF2C only activates CS-1. CONCLUSION: Our present data suggest that NF-κB is required for the transcription of mouse CS-1 but not CS-2, and that the regulation of the calsarcins is mediated also by the NFAT and MEF2 transcription factors. These results provide new insights into the molecular mechanisms governing transcription in specific muscle fibre cells. The calsarcins may also serve as a valuable mechanistic tool to better understand the regulation of calcineurin signalling during muscle differentiation