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

Analysis of the current situation and trends in research on Kunqu opera costumes based on the China National Knowledge Infrastructure

The research on Kunqu opera costumes presents a diversified and multidimensional trend in the 21st century. This article analyzes the current situation of research on Kunqu opera costumes in China by examining the annual distribution of literature on the subject on the Chinese Knowledge Network, as well as the keywords and fluctuations in research areas. The aim is to explore the hot issues and trends in research on Kunqu opera costumes in China in recent years, in order to provide support for the protection and development of traditional theater. In the future, the focus of research will be on innovative design of Kunqu opera costumes, cultural heritage, digital technology, and multicultural studies

Reconfigurable resonant and leaky-wave antennas for future wireless systems

University of Technology Sydney. Faculty of Engineering and Information Technology.Reconfigurable antennas (RAs), which have agilities in their frequency and radiation characteristics, are emerging as one of the key enabling components for wireless systems. Compared with single-function legacy antennas, RAs can greatly enhance the system functionalities and flexibilities. In this dissertation, on the basis of state-of-the-art of RAs, a thorough investigation, development, and implementation are conducted on reconfigurable resonant and leaky-wave antennas to fully exploit and extend the benefits of RAs. In order to enable advanced technologies for reconfigurable resonant antennas, challenging and promising gaps associated with novel designs are addressed on polarization-reconfigurable antenna, pattern-reconfigurable antenna, as well as combined polarization- and pattern-reconfigurable antenna. In particular, a reconfigurable center-fed shorting-via loaded patch antenna is designed that can switch among four linear polarizations at a 45° rotation. Furthermore, by reconfiguring parasitic striplines and bottom reflecting metal pieces around a radiating dipole, five switchable beams including broadside, titled, and endfire directions, are achieved. Finally, a cavity-backed proximity-coupled patch antenna operating at 11 GHz is designed and reconfigured to switch among three linear polarizations, and its main beam can be steered to about +20°, 0°, -20° with a measured gain variation between 7.2 and 8.1 dBi for each polarization. As one of the most widely used types of non-resonant antennas, leaky-wave antennas (LWAs) serially leak power through open traveling-wave structures, and they generally have inherent beam scanning capability by sweeping the source frequency. One main challenge in LWAs is to realize a wide-angle continuous circular-polarization (CP) beam scan. A novel benzene-ring-shaped slot-loaded CP substrate-integrated-waveguide LWA with partially reflecting wall vias is developed herein to scan a CP beam from -42.8° to +54.3° with a gain variation of only 3.3 dBic. Moreover, a polarization reconfigurable LWA that can switch between its circular-polarization and linear-polarization states is investigated and accomplished to further improve the diversity and functionality delivered by the LWA system. Finally, to eliminate the frequency-dependant beam scanning behaviour, a reconfigurable composite right/left-handed (CRLH) LWA is developed that facilitates fixed-frequency continuous beam scanning over a wide operational frequency band. Since the wireless systems are evolving towards multi-functionality to enable fast, secure, and reliable connections, it is believed that the developed reconfigurable resonant and leaky-wave antennas can find wide applications in current and beyond wireless systems

Preparation and adsorbing sodium borohydride of porous hollow capsules

AbstractSodium borohydride, a solid material, has great attractive for its hydrogen storage and preparation properties. The solubility of sodium borohydride in water is up to 35% at room temperature. It can react with water for generating hydrogen. The merits of this reaction include high purity, mild reaction conditions and the high theoretical density of hydrogen generation. Complete hydrolysis of sodium borohydride can produce hydrogen and sodium metaborate, which can be recovered by advanced technology for sodium borohydride recycling. Porous hollow capsules containing nickel boride were prepared and used as storage and reaction space for sodium borohydride. The influences of the concentration of polymer solution, the ratio of the coagulation bath, the concentration and temperature on the porous structure of hollow capsule were investigated. The adsorption of porous hollow capsule was influenced and optimized by soaking time, adsorption conditions, the drying temperature and time. The best conditions of preparation of porous hollow capsule are: 15 wt% PVDF into capsule system configuration, with adding 15wt % attapulgite or 5 wt% PVP. The adsorption amount is up to 36%. The preparation method of porous hollow capsule is simple and easy to operate, low energy consumption, simple process only including dissolving, mixing, molding, adsorption and drying. The structure of porous hollow is stable and easy to storage and use. Hydrogen can be simple to release when mixed the adsorbed capsules with water