Uniform Asymptotics of Orthogonal Polynomials Arising from Coherent States

In this paper, we study a family of orthogonal polynomials $\{\phi_n(z)\}$ arising from nonlinear coherent states in quantum optics. Based on the three-term recurrence relation only, we obtain a uniform asymptotic expansion of $\phi_n(z)$ as the polynomial degree $n$ tends to infinity. Our asymptotic results suggest that the weight function associated with the polynomials has an unusual singularity, which has never appeared for orthogonal polynomials in the Askey scheme. Our main technique is the Wang and Wong's difference equation method. In addition, the limiting zero distribution of the polynomials $\phi_n(z)$ is provided

Acoustic Emission Detection of Early Stages of Cracks in Rotating Gearbox Components

Many critical, highly loaded rotating gearbox components have fast crack propagation rates. Early detection of cracks in gearbox is critical to mitigating the risk of catastrophic failure. Acoustic Emission (AE) techniques have proven to be capable of continuously monitoring the crack initiation and propagation [1-3]. Due to the long distance of AE signal propagation from the AE sources to the sensors installed in the housing, the AE signal suffers from severe attenuation and noises. Accurate AE signal classification technology that is capable of extracting the true AE signal out of background noises generated by the surrounding environment of a gearbox is desired. In this paper, an innovative feature extraction and analysis based AE signal classification technology is developed to address this issue. Potential AE signals are first pulled out of the noisy background in real-time through a set of automated AE detection algorithms. Then features including count, energy, duration, amplitude, rise time, amplitude rise time ratio, etc. are extracted and analyzed. Through the comparison and correlation of features extracted from signals recorded by multiple AE sensors, respective feature thresholds are determined to distinguish noises from real AE signal. The classification results are experimentally validated through fatigue tests

Plancherel-Rotach asymptotic expansion for some polynomials from indeterminate moment problems

We study the Plancherel--Rotach asymptotics of four families of orthogonal polynomials, the Chen--Ismail polynomials, the Berg-Letessier-Valent polynomials, the Conrad--Flajolet polynomials I and II. All these polynomials arise in indeterminate moment problems and three of them are birth and death process polynomials with cubic or quartic rates. We employ a difference equation asymptotic technique due to Z. Wang and R. Wong. Our analysis leads to a conjecture about large degree behavior of polynomials orthogonal with respect to solutions of indeterminate moment problems.Comment: 34 pages, typos corrected and references update