Hankel Transform in Quantum Calculus and Applications

Mathematics Subject Classification: 44A15, 33D15, 81Q99

On Hankel Transform of Generalized Mathieu Series

Mathematics Subject Classification: Primary 33E20, 44A10; Secondary 33C10, 33C20, 44A20

CR1Dmod: A Matlab program to model 1D complex resistivity effects in electrical and electromagnetic surveys

We have constructed a forward modelling code in Matlab, capable of handling several commonly used electrical and electromagnetic methods in a 1D environment. We review the implemented electromagnetic field equations for grounded wires, frequency and transient soundings and present new solutions in the case of a non-magnetic first layer. The CR1Dmod code evaluates the Hankel transforms occurring in the field equations using either the Fast Hankel Transform based on digital filter theory, or a numerical integration scheme applied between the zeros of the Bessel function. A graphical user interface allows easy construction of 1D models and control of the parameters. Modelling results are in agreement with other authors, but the time of computation is less efficient than other available codes. Nevertheless, the CR1Dmod routine handles complex resistivities and offers solutions based on the full EM-equations as well as the quasi-static approximation. Thus, modelling of effects based on changes in the magnetic permeability and the permittivity is also possible.

Exercise sheet 5

Exercises and solutions in LaTex

Exercise sheet 5 pdf

Exercises and solutions in PDF

Desenvolvimento de antenas de microfita com Patch em anel utilizando materiais ferrimagnéticos e metamateriais

In general, the materials used as substrates in the project of microstrip antennas are: isotropic, anisotropic dielectrics and ferrimagnetic materials (magnetic anisotropy). The use of ferrimagnetic materials as substrates in microstrip patch antennas has been concentrated on the analysis of antennas with circular and rectangular patches. However, a new class of materials, called metamaterials, has been currently the focus of a great deal of interest. These materials exhibit bianisotropic characteristics, with permittivity and permeability tensors. The main objective of this work is to develop a theoretical and numerical analysis for the radiation characteristics of annular ring microstrip antennas, using ferrites and metamaterials as substrates. The full wave analysis is performed in the Hankel transform domain through the application of the Hertz vector potentials. Considering the definition of the Hertz potentials and imposing the boundary conditions, the dyadic Green s function components are obtained relating the surface current density components at the plane of the patch to the electric field tangential components. Then, Galerkin s method is used to obtain a system of matrix equations, whose solution gives the antenna resonant frequency. From this modeling, it is possible to obtain numerical results for the resonant frequency, radiation pattern, return loss, and antenna bandwidth as a function of the annular ring physical parameters, for different configurations and substrates. The theoretical analysis was developed for annular ring microstrip antennas on a double ferrimagnetic/isotropic dielectric substrate or metamaterial/isotropic dielectric substrate. Also, the analysis for annular ring microstrip antennas on a single ferrimagnetic or metamaterial layer and for suspended antennas can be performed as particular cases

Antenas de microfita com patch em anel e múltiplas camadas dielétricas anisotrópicas uniaxiais

This work presents an analysis of the annular ring microstrip antennas printed on uniaxial anisotropic substrates and with superstrate.The analysis uses the full-wave formulation by means of the Hertz vector potentials method, in the Hankel transform domain. The definition of the Hertz vector potentials and the application of the appropriate boundary conditions to the structure allow determining the dyadic Green functions, relating the current densities in the conducting patch to the transforms of the tangential electric field components. Galerkin s method is then used to obtain the matrix equation whose nontrivial solution gives the complex resonant frequency of the antenna. From the modeling, it is possible to obtain results for the resonant frequency, bandwidth and quality factor, as a function of several parameters of the antenna, for different configurations. We have considered annular ring microstrip antennas on a single dielectric layer, antennas with two anisotropic dielectric layers, and annular ring microstrip antennas on suspended substrates. Numerical results for the resonant frequency of the these structures printed on isotropic substrates are also presented and compared with those published by other authors, showing a good agreement

Análise de antenas de microfita com patches circulares sobre substratos anisotrópicos usando o método dos potenciais de Hertz

This work consists on the theoretical and numerical analysis of some properties of circular microstrip patch antennas on isotropic and uniaxial anisotropic substrates. For this purpose, a full wave analysis is performed, using Hertz Vector Potentials method in the Hankel Transform domain. In the numerical analysis, the moment method is also used in order to determine some characteristics of the antenna, such as: resonant frequency and radiation pattern. The definition of Hertz potentials in the Hankel domain is used in association with Maxwell´s equations and the boundary conditions of the structures to obtain the Green´s functions, relating the components of the current density on the patch and the tangential electric field components. Then, the Galerkin method is used to generate a matrix equation whose nontrivial solution is the complex resonant frequency of the structure. In the analysis, a microstrip antenna with only one isotropic dielectric layer is initially considered. For this structure, the effect of using superconductor patches is also analyzed. An analysis of a circular microstrip antenna on an uniaxial anisotropic dielectric layer is performed, using the Hertz vector potentials oriented along the optical axis of the material, that is perpendicular to the microstrip ground plane. Afterwards, the circular microstrip antenna using two uniaxial anisotropic dielectric layers is investigated, considering the particular case in which the inferior layer is filled by air. In this study, numerical results for resonant frequency and radiation pattern for circular microstrip antennas on isotropic and uniaxial anisotropic substrates are presented and compared with measured and calculated results found in the literature

Hankel operators on Fock spaces

We study Hankel operators on the weighted Fock spaces Fp. The boundedness and compactness of these operators are characterized in terms of BMO and VMO, respectively. Along the way, we also study Berezin transform and harmonic conjugates on the plane. Our results are analogous to Zhu's characterization of bounded and compact Hankel operators on Bergman spaces of the unit disk.

Generalized Convolution Transforms and Toeplitz Plus Hankel Integral Equations

Mathematics Subject Classification: 44A05, 44A35

Analysis Of The Dna Fourier Transform-infrared Microspectroscopic Signature Using An All-reflecting Objective.

The Fourier transform-infrared (FT-IR) signature of dry samples of DNA and DNA-polypeptide complexes, as studied by IR microspectroscopy using a diamond attenuated total reflection (ATR) objective, has revealed important discriminatory characteristics relative to the PO2(-) vibrational stretchings. However, DNA IR marks that provide information on the sample's richness in hydrogen bonds have not been resolved in the spectral profiles obtained with this objective. Here we investigated the performance of an all reflecting objective (ARO) for analysis of the FT-IR signal of hydrogen bonds in DNA samples differing in base richness types (salmon testis vs calf thymus). The results obtained using the ARO indicate prominent band peaks at the spectral region representative of the vibration of nitrogenous base hydrogen bonds and of NH and NH2 groups. The band areas at this spectral region differ in agreement with the DNA base richness type when using the ARO. A peak assigned to adenine was more evident in the AT-rich salmon DNA using either the ARO or the ATR objective. It is concluded that, for the discrimination of DNA IR hydrogen bond vibrations associated with varying base type proportions, the use of an ARO is recommended.

POSITIVITY PROPERTIES OF THE FOURIER TRANSFORM AND THE STABILITY OF PERIODIC TRAVELLING-WAVE SOLUTIONS

In this paper we establish a method to obtain the stability of periodic travelling-wave solutions for equations of Korteweg-de Vries-type u(t) + u(p)u(x) - Mu(x) = 0, with M being a general pseudodifferential operator and where p >= 1 is an integer. Our approach uses the theory of totally positive operators, the Poisson summation theorem, and the theory of Jacobi elliptic functions. In particular we obtain the stability of a family of periodic travelling waves solutions for the Benjamin Ono equation. The present technique gives a new way to obtain the existence and stability of cnoidal and dnoidal waves solutions associated with the Korteweg-de Vries and modified Korteweg-de Vries equations, respectively. The theory has prospects for the study of periodic travelling-wave solutions of other partial differential equations.

Estimation of hexenuronic acids and kappa number in kraft pulps of Eucalyptus globulus by Fourier transform near infrared spectroscopy and multivariate analysis

Fourier transform near infrared (FT-NIR) spectroscopy was evaluated as an analytical too[ for monitoring residual Lignin, kappa number and hexenuronic acids (HexA) content in kraft pulps of Eucalyptus globulus. Sets of pulp samples were prepared under different cooking conditions to obtain a wide range of compound concentrations that were characterised by conventional wet chemistry analytical methods. The sample group was also analysed using FT-NIR spectroscopy in order to establish prediction models for the pulp characteristics. Several models were applied to correlate chemical composition in samples with the NIR spectral data by means of PCR or PLS algorithms. Calibration curves were built by using all the spectral data or selected regions. Best calibration models for the quantification of lignin, kappa and HexA were proposed presenting R-2 values of 0.99. Calibration models were used to predict pulp titers of 20 external samples in a validation set. The lignin concentration and kappa number in the range of 1.4-18% and 8-62, respectively, were predicted fairly accurately (standard error of prediction, SEP 1.1% for lignin and 2.9 for kappa). The HexA concentration (range of 5-71 mmol kg(-1) pulp) was more difficult to predict and the SEP was 7.0 mmol kg(-1) pulp in a model of HexA quantified by an ultraviolet (UV) technique and 6.1 mmol kg(-1) pulp in a model of HexA quantified by anion-exchange chromatography (AEC). Even in wet chemical procedures used for HexA determination, there is no good agreement between methods as demonstrated by the UV and AEC methods described in the present work. NIR spectroscopy did provide a rapid estimate of HexA content in kraft pulps prepared in routine cooking experiments.

Power Transformer Differential Protection Based on Clarke`s Transform and Fuzzy Systems

The power transformer is a piece of electrical equipment that needs continuous monitoring and fast protection since it is very expensive and an essential element for a power system to perform effectively. The most common protection technique used is the percentage differential logic, which provides discrimination between an internal fault and different operating conditions. Unfortunately, there are some operating conditions of power transformers that can affect the protection behavior and the power system stability. This paper proposes the development of a new algorithm to improve the differential protection performance by using fuzzy logic and Clarke`s transform. An electrical power system was modeled using Alternative Transients Program (ATP) software to obtain the operational conditions and fault situations needed to test the algorithm developed. The results were compared to a commercial relay for validation, showing the advantages of the new method.

Identification of protein coding regions using the modified Gabor-wavelet transform

An important topic in genomic sequence analysis is the identification of protein coding regions. In this context, several coding DNA model-independent methods based on the occurrence of specific patterns of nucleotides at coding regions have been proposed. Nonetheless, these methods have not been completely suitable due to their dependence on an empirically predefined window length required for a local analysis of a DNA region. We introduce a method based on a modified Gabor-wavelet transform (MGWT) for the identification of protein coding regions. This novel transform is tuned to analyze periodic signal components and presents the advantage of being independent of the window length. We compared the performance of the MGWT with other methods by using eukaryote data sets. The results show that MGWT outperforms all assessed model-independent methods with respect to identification accuracy. These results indicate that the source of at least part of the identification errors produced by the previous methods is the fixed working scale. The new method not only avoids this source of errors but also makes a tool available for detailed exploration of the nucleotide occurrence.