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Studies of equivalent fuzzy subgroups of finite abelian p-Groups of rank two and their subgroup lattices

**Authors:**Ngcibi, Sakhile Leonard**Date:**2006**Subjects:**Abelian groups , Fuzzy sets , Finite groups , Group theory , Polynomials**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5416 , http://hdl.handle.net/10962/d1005230**Description:**We determine the number and nature of distinct equivalence classes of fuzzy subgroups of finite Abelian p-group G of rank two under a natural equivalence relation on fuzzy subgroups. Our discussions embrace the necessary theory from groups with special emphasis on finite p-groups as a step towards the classification of crisp subgroups as well as maximal chains of subgroups. Unique naming of subgroup generators as discussed in this work facilitates counting of subgroups and chains of subgroups from subgroup lattices of the groups. We cover aspects of fuzzy theory including fuzzy (homo-) isomorphism together with operations on fuzzy subgroups. The equivalence characterization as discussed here is finer than isomorphism. We introduce the theory of keychains with a view towards the enumeration of maximal chains as well as fuzzy subgroups under the equivalence relation mentioned above. We discuss a strategy to develop subgroup lattices of the groups used in the discussion, and give examples for specific cases of prime p and positive integers n,m. We derive formulas for both the number of maximal chains as well as the number of distinct equivalence classes of fuzzy subgroups. The results are in the form of polynomials in p (known in the literature as Hall polynomials) with combinatorial coefficients. Finally we give a brief investigation of the results from a graph-theoretic point of view. We view the subgroup lattices of these groups as simple, connected, symmetric graphs.**Full Text:**

**Authors:**Ngcibi, Sakhile Leonard**Date:**2006**Subjects:**Abelian groups , Fuzzy sets , Finite groups , Group theory , Polynomials**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5416 , http://hdl.handle.net/10962/d1005230**Description:**We determine the number and nature of distinct equivalence classes of fuzzy subgroups of finite Abelian p-group G of rank two under a natural equivalence relation on fuzzy subgroups. Our discussions embrace the necessary theory from groups with special emphasis on finite p-groups as a step towards the classification of crisp subgroups as well as maximal chains of subgroups. Unique naming of subgroup generators as discussed in this work facilitates counting of subgroups and chains of subgroups from subgroup lattices of the groups. We cover aspects of fuzzy theory including fuzzy (homo-) isomorphism together with operations on fuzzy subgroups. The equivalence characterization as discussed here is finer than isomorphism. We introduce the theory of keychains with a view towards the enumeration of maximal chains as well as fuzzy subgroups under the equivalence relation mentioned above. We discuss a strategy to develop subgroup lattices of the groups used in the discussion, and give examples for specific cases of prime p and positive integers n,m. We derive formulas for both the number of maximal chains as well as the number of distinct equivalence classes of fuzzy subgroups. The results are in the form of polynomials in p (known in the literature as Hall polynomials) with combinatorial coefficients. Finally we give a brief investigation of the results from a graph-theoretic point of view. We view the subgroup lattices of these groups as simple, connected, symmetric graphs.**Full Text:**

Modelling and investigating primary beam effects of reflector antenna arrays

**Authors:**Iheanetu, Kelachukwu**Date:**2020**Subjects:**Antennas, Reflector , Radio telescopes , Astronomical instruments -- Calibration , Holography , Polynomials , Very large array telescopes -- South Africa , Astronomy -- Data processing , Primary beam effects , Jacobi-Bessel pattern , Cassbeam software , MeerKAT telescope**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**http://hdl.handle.net/10962/147425 , vital:38635**Description:**Signals received by a radio telescope are always affected by propagation and instrumental effects. These effects need to be modelled and accounted for during the process of calibration. The primary beam (PB) of the antenna is one major instrumental effect that needs to be accounted for during calibration. Producing accurate models of the radio antenna PB is crucial, and many approaches (like electromagnetic and optical simulations) have been used to model it. The cos³ function, Jacobi-Bessel pattern, characteristic basis function patterns (CBFP) and Cassbeam software (which uses optical ray-tracing with antenna parameters) have also been used to model it. These models capture the basic PB effects. Real-life PB patterns differ from these models due to various subtle effects such as mechanical deformation and effects introduced into the PB due to standing waves that exist in reflector antennas. The actual patterns can be measured via a process called astro-holography (or holography), but this is subject to noise, radio frequency interference, and other measurement errors. In our approach, we use principal component analysis and Zernike polynomials to model the PBs of the Very Large Array (VLA) and the MeerKAT telescopes from their holography measured data. The models have reconstruction errors of less than 5% at a compression factor of approximately 98% for both arrays. We also present steps that can be used to generate accurate beam models for any telescope (independent of its design) based on holography measured data. Analysis of the VLA measured PBs revealed that the graph of the beam sizes (and centre offset positions) have a fast oscillating trend (superimposed on a slow trend) with frequency. This spectral behaviour we termed ripple or characteristic effects. Most existing PB models that are used in calibrating VLA data do not incorporate these direction dependent effects (DDEs). We investigate the impact of using PB models that ignore this DDE in continuum calibration and imaging via simulations. Our experiments show that, although these effects translate into less than 10% errors in source flux recovery, they do lead to 30% reduction in the dynamic range. To prepare data for Hi and radio halo (faint emissions) science analysis requires carrying out foreground subtraction of bright (continuum) sources. We investigate the impact of using beam models that ignore these ripple effects during continuum subtraction. These show that using PB models which completely ignore the ripple effects in continuum subtraction could translate to error of more to 30% in the recovered Hi spectral properties. This implies that science inferences drawn from the results for Hi studies could have errors of the same magnitude.**Full Text:**

**Authors:**Iheanetu, Kelachukwu**Date:**2020**Subjects:**Antennas, Reflector , Radio telescopes , Astronomical instruments -- Calibration , Holography , Polynomials , Very large array telescopes -- South Africa , Astronomy -- Data processing , Primary beam effects , Jacobi-Bessel pattern , Cassbeam software , MeerKAT telescope**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**http://hdl.handle.net/10962/147425 , vital:38635**Description:**Signals received by a radio telescope are always affected by propagation and instrumental effects. These effects need to be modelled and accounted for during the process of calibration. The primary beam (PB) of the antenna is one major instrumental effect that needs to be accounted for during calibration. Producing accurate models of the radio antenna PB is crucial, and many approaches (like electromagnetic and optical simulations) have been used to model it. The cos³ function, Jacobi-Bessel pattern, characteristic basis function patterns (CBFP) and Cassbeam software (which uses optical ray-tracing with antenna parameters) have also been used to model it. These models capture the basic PB effects. Real-life PB patterns differ from these models due to various subtle effects such as mechanical deformation and effects introduced into the PB due to standing waves that exist in reflector antennas. The actual patterns can be measured via a process called astro-holography (or holography), but this is subject to noise, radio frequency interference, and other measurement errors. In our approach, we use principal component analysis and Zernike polynomials to model the PBs of the Very Large Array (VLA) and the MeerKAT telescopes from their holography measured data. The models have reconstruction errors of less than 5% at a compression factor of approximately 98% for both arrays. We also present steps that can be used to generate accurate beam models for any telescope (independent of its design) based on holography measured data. Analysis of the VLA measured PBs revealed that the graph of the beam sizes (and centre offset positions) have a fast oscillating trend (superimposed on a slow trend) with frequency. This spectral behaviour we termed ripple or characteristic effects. Most existing PB models that are used in calibrating VLA data do not incorporate these direction dependent effects (DDEs). We investigate the impact of using PB models that ignore this DDE in continuum calibration and imaging via simulations. Our experiments show that, although these effects translate into less than 10% errors in source flux recovery, they do lead to 30% reduction in the dynamic range. To prepare data for Hi and radio halo (faint emissions) science analysis requires carrying out foreground subtraction of bright (continuum) sources. We investigate the impact of using beam models that ignore these ripple effects during continuum subtraction. These show that using PB models which completely ignore the ripple effects in continuum subtraction could translate to error of more to 30% in the recovered Hi spectral properties. This implies that science inferences drawn from the results for Hi studies could have errors of the same magnitude.**Full Text:**

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