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

Gaussian Unitary Ensembles with Fisher-Hartwig Singularities and Generalized Painlev\'{e} IV Equation

We study the Hankel determinant generated by the Gaussian weight with Fisher-Hartwig singularities of root type at $t_j$, $j=1,\cdots ,N$. We derive the ladder operators satisfied by the associated monic orthogonal polynomials and three compatibility conditions. By using them and introducing $2N$ auxiliary quantities $\{R_{n,j}, r_{n,j}, j=1,\cdots,N\}$, we build a series of difference equations. Furthermore, we prove that $\{R_{n,j}, r_{n,j}\}$ satisfy Riccati equations. From them we deduce a system of second order PDEs satisfied by $\{R_{n,j}, j=1,\cdots,N\}$, which is reduced to a Painlev\'{e} IV equation for $N=1$. We also show that the logarithmic derivative of the Hankel determinant satisfies the generalized $\sigma$-form of a Painlev\'{e} IV equation

Effect of Ultrasonic Vibration on Microstructure and Mechanical Properties of Nano-SiC_p/Al-5Cu Composites

The ultrasonic vibration (UV) treatment has been successfully applied to improve the particles distribution of nano-SiCp reinforced Al-5Cu composites fabricated by combined processes of dry high energy ball milling and squeeze casting. Large particles aggregates are eliminated by effects of the cavitation and the acoustic streaming of UV for 1 min. All the particles aggregates are eliminated and the particles are uniformly distributed in the melt after treated by UV for 5 min. The ultimate tensile strength, yield strength and elongation of the 1%(mass fraction) nano-sized SiCp/Al-5Cu composites treated by UV for 5 min are 270 MPa, 173 MPa and 13.3%, which are increased by 7.6%, 6.8% and 29% compared with Al-5Cu matrix alloy, respectively, and are increased by 27.4%, 9.5% and 303% compared with the untreated composites, respectively. The improvements of mechanical properties after UV are attributed to the uniform distribution of reinforcement and grain refinement of aluminum alloy matrix

Cauliflower mosaic virus P6 Dysfunctions Histone Deacetylase HD2C to Promote Virus Infection

Histone deacetylases (HDACs) are vital epigenetic modifiers not only in regulating plant development but also in abiotic- and biotic-stress responses. Though to date, the functions of HD2C—an HD2-type HDAC—In plant development and abiotic stress have been intensively explored, its function in biotic stress remains unknown. In this study, we have identified HD2C as an interaction partner of the Cauliflower mosaic virus (CaMV) P6 protein. It functions as a positive regulator in defending against CaMV infection. The hd2c mutants show enhanced susceptibility to CaMV infection. In support, the accumulation of viral DNA, viral transcripts, and the deposition of histone acetylation on the viral minichromosomes are increased in hd2c mutants. P6 interferes with the interaction between HD2C and HDA6, and P6 overexpression lines have similar phenotypes with hd2c mutants. In further investigations, P6 overexpression lines, together with CaMV infection plants, are more sensitive to ABA and NaCl with a concomitant increasing expression of ABA/NaCl-regulated genes. Moreover, the global levels of histone acetylation are increased in P6 overexpression lines and CaMV infection plants. Collectively, our results suggest that P6 dysfunctions histone deacetylase HD2C by physical interaction to promote CaMV infection

Genomic analysis based on chromosome-level genome assembly reveals Myrtaceae evolution and terpene biosynthesis of rose myrtle

Abstract Background Rose myrtle (Rhodomyrtus tomentosa (Ait.) Hassk), is an evergreen shrub species belonging to the family Myrtaceae, which is enriched with bioactive volatiles (α-pinene and β-caryophyllene) with medicinal and industrial applications. However, the mechanism underlying the volatile accumulation in the rose myrtle is still unclear. Results Here, we present a chromosome-level genomic assembly of rose myrtle (genome size = 466 Mb, scaffold N50 = 43.7 Mb) with 35,554 protein-coding genes predicted. Through comparative genomic analysis, we found that gene expansion and duplication had a potential contribution to the accumulation of volatile substances. We proposed that the action of positive selection was significantly involved in volatile accumulation. We identified 43 TPS genes in R. tomentosa. Further transcriptomic and TPS gene family analyses demonstrated that the distinct gene subgroups of TPS may contribute greatly to the biosynthesis and accumulation of different volatiles in the Myrtle family of shrubs and trees. The results suggested that the diversity of TPS-a subgroups led to the accumulation of special sesquiterpenes in different plants of the Myrtaceae family. Conclusions The high quality chromosome-level rose myrtle genome and the comparative analysis of TPS gene family open new avenues for obtaining a higher commercial value of essential oils in medical plants