Complex Chebyshev polynomials and generalizations with an application to the optimal choice of interpolating knots in complex planar splines

AbstractThis paper contains a brief account on complex planar splines which are complex valued functions defined piecewise on a grid. For noncontinuous (so called nonconforming) splines the problem of the placement of knots at which these splines are required to be continuous is investigated. It is shown that this problem reduces to finding complex Chebyshev polynomials under the additional requirement that the zeros of the polynomials are on the boundary of the corresponding domains. It is proved that the zeros of a generalized Chebyshev polynomial are in the convex hull of the domain on which the Chebyshev polynomials are defined. Some open problems are stated. A numerical and graphical display for the optimal location of three and six points on certain triangles is provided

On the rate of convergence in the central limit theorem for signed rank statistics

AbstractThe problem of the rate of convergence of the cumulative distribution function of the one-sample rank order statistics SN+ to the limiting normal one is studied for the case when the underlying observations are independent but not necessarily identically distributed. The results obtained are then used to derive a strong law and functional form of a central limit theorem for SN

Local maxima of the sample functions of the N-parameter Bessel process

AbstractIn this paper we show that almost every sample function of the N-parameter Bessel process associated with the N-parameter Wiener process has a local maximum. In addition some properties related to the local maxima are investigated

Nonparametric Approach for Non-Gaussian Vector Stationary Processes

AbstractSuppose that {z(t)} is a non-Gaussian vector stationary process with spectral density matrixf(λ). In this paper we consider the testing problemH:∫π−πK{f(λ)}dλ=cagainstA:∫π−πK{f(λ)}dλ≠c, whereK{·} is an appropriate function andcis a given constant. For this problem we propose a testTnbased on ∫π−πK{f(λ)}dλ=c, wheref(λ) is a nonparametric spectral estimator off(λ), and we define an efficacy ofTnunder a sequence of nonparametric contiguous alternatives. The efficacy usually depnds on the fourth-order cumulant spectraf4Zofz(t). If it does not depend onf4Z, we say thatTnis non-Gaussian robust. We will give sufficient conditions forTnto be non-Gaussian robust. Since our test setting is very wide we can apply the result to many problems in time series. We discuss interrelation analysis of the components of {z(t)} and eigenvalue analysis off(λ). The essential point of our approach is that we do not assume the parametric form off(λ). Also some numerical studies are given and they confirm the theoretical results

Actuarial and Financial Risks: Models, Statistical Inference, and Case Studies

Publisher's, offprint versionUnderstanding actuarial and financial risks poses major challenges. The need for reliable approaches to risk assessment is particularly acute in the present context of highly uncertain financial markets. New regulatory guidelines such as the Basel II Accord for banking and Solvency II for insurance are being implemented in many parts of the world. Regulators in various countries are adopting risk-based approaches to the supervision of financial institutions

Review of Graphene Technology and Its Applications for Electronic Devices

Graphene has amazing abilities due to its unique band structure characteristics defining its enhanced electrical capabilities for a material with the highest characteristic mobility known to exist at room temperature. The high mobility of graphene occurs due to electron delocalization and weak electron–phonon interaction, making graphene an ideal material for electrical applications requiring high mobility and fast response times. In this review, we cover graphene’s integration into infrared (IR) devices, electro-optic (EO) devices, and field effect transistors (FETs) for radio frequency (RF) applications. The benefits of utilizing graphene for each case are discussed, along with examples showing the current state-of-the-art solutions for these applications

Enzyme immobilisation on amino-functionalised multi-walled carbon nanotubes : structural and biocatalytic characterisation

BACKGROUND: The aim of this work is to investigate the structure and function of enzymes immobilised on nanomaterials. This work will allow better understanding of enzyme-nanomaterial interactions, as well as designing functional protein-nanomaterial conjugates. METHODOLOGY/PRINCIPAL FINDINGS: Multiwalled carbon nanotubes (MWNTs) were functionalised with amino groups to improve solubility and biocompatibility. The pristine and functionalised forms of MWNTs were characterised with Fourier-transform infrared spectroscopy. Thermogravimetric analysis was done to examine the degree of the functionalisation process. An immobilised biocatalyst was prepared on functionalised nanomaterial by covalent binding. Thermomyces lanuginosus lipase was used as a model enzyme. The structural change of the immobilised and free lipases were characterised with transmission electron Microscopy, X-ray photoelectron spectroscopy, Fourier-transform infrared spectroscopy and Circular dichroism spectroscopy. Biochemical characterisation of immobilised enzyme showed broader pH and thermal optima compared to soluble form. Reusability of the immobilised enzyme for hydrolysis of long chain esters was demonstrated up to ten cycles. CONCLUSION/SIGNIFICANCE: Lipase immobilised on MWNTs has exhibited significantly improved thermal stability. The exploration of advanced nanomaterial for enzyme immobilisation support using sophisticated techniques makes nanobiocatalyst of potential interest for biosensor applications

Nonparametric density estimators based on nonstationary absolutely regular random sequences

In this paper, the central limit theorems for the density estimator and for the integrated square error are proved for the case when the underlying sequence of random variables is nonstationary. Applications to Markov processes and ARMA processes are provided