An investigation of the efficacy of the Office of the Auditor-General in ensuring public financial accountability in the Gauteng Province
- Authors: Khumalo, Mildred Tariro
- Date: 2007
- Subjects: Finance, Public -- South Africa , Budget -- South Africa -- Gauteng , Liability (Law)
- Language: English
- Type: Thesis , Masters , MPA
- Identifier: vital:11645 , http://hdl.handle.net/10353/283 , Finance, Public -- South Africa , Budget -- South Africa -- Gauteng , Liability (Law)
- Description: Increasing accountability is a key element in a wide variety of governance reforms, from national level, public sector restructuring, to anti corruption campaigns and to centralise service delivery at local levels. (Van Wyk, 2002:311). Financial accountability has a major effect on performance issues, and these two combined have implications on democratic accountability. It is obviously a fact that when public officials become accountable, this will have a positive effect on service delivery. The introduction of the Public Finance Management Act (PFMA) 1999 (Act 1 of 1999) as amended by Act 29 of 1999 adopts an approach to financial management, which emphasises the need for accountability of results by focusing on output and responsibility rather than the rule driven approach of the previous Exchequer Acts. One quickly realises that the PFMA aims to regulate financial management at both National and Provincial levels of government. It directs attention at revenue, expenditure, assets and liabilities and aims to see that these are managed efficiently and effectively. Public finance accountability is increasingly becoming important in the public sector. One means of effecting accountability is through auditing. For this reason the Office of the Auditor -General through the Constitution of the Republic of South Africa 1996 (Act 108 of 1996 section 216 (1)) was established to facilitate effective accountability through auditing. The Office of the Auditor -General’s role is by no means minor, for it plays a major role in curbing corruption and acts as a ‘watchdog’ of the nation over public funds. Amidst high mismanagement and misuse of public resources, as well as corruption, especially in developing countries, the major question one asks is how effective the Auditor -General can be in ensuring effective public financial management and accountability within state departments so as to improve service delivery.
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- Authors: Khumalo, Mildred Tariro
- Date: 2007
- Subjects: Finance, Public -- South Africa , Budget -- South Africa -- Gauteng , Liability (Law)
- Language: English
- Type: Thesis , Masters , MPA
- Identifier: vital:11645 , http://hdl.handle.net/10353/283 , Finance, Public -- South Africa , Budget -- South Africa -- Gauteng , Liability (Law)
- Description: Increasing accountability is a key element in a wide variety of governance reforms, from national level, public sector restructuring, to anti corruption campaigns and to centralise service delivery at local levels. (Van Wyk, 2002:311). Financial accountability has a major effect on performance issues, and these two combined have implications on democratic accountability. It is obviously a fact that when public officials become accountable, this will have a positive effect on service delivery. The introduction of the Public Finance Management Act (PFMA) 1999 (Act 1 of 1999) as amended by Act 29 of 1999 adopts an approach to financial management, which emphasises the need for accountability of results by focusing on output and responsibility rather than the rule driven approach of the previous Exchequer Acts. One quickly realises that the PFMA aims to regulate financial management at both National and Provincial levels of government. It directs attention at revenue, expenditure, assets and liabilities and aims to see that these are managed efficiently and effectively. Public finance accountability is increasingly becoming important in the public sector. One means of effecting accountability is through auditing. For this reason the Office of the Auditor -General through the Constitution of the Republic of South Africa 1996 (Act 108 of 1996 section 216 (1)) was established to facilitate effective accountability through auditing. The Office of the Auditor -General’s role is by no means minor, for it plays a major role in curbing corruption and acts as a ‘watchdog’ of the nation over public funds. Amidst high mismanagement and misuse of public resources, as well as corruption, especially in developing countries, the major question one asks is how effective the Auditor -General can be in ensuring effective public financial management and accountability within state departments so as to improve service delivery.
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Assessment of antibacterial potentials of Garcinia Kola seed extracts and their interactions with antibiotics
- Authors: Sibanda, Thulani
- Date: 2007
- Subjects: Drug resistance in microorganisms , Garcinia , Antibiotics , Medicinal plants
- Language: English
- Type: Thesis , Masters , MSc (Microbiology)
- Identifier: vital:11242 , http://hdl.handle.net/10353/71 , Drug resistance in microorganisms , Garcinia , Antibiotics , Medicinal plants
- Description: The antibacterial potency of the extracts of the seed of Garcinia kola (bitter kola) was investigated in this study against a panel of referenced, environmental and clinical bacterial strains. The killing rates of the active extract as well as their potential for combination antibacterial therapy with standard antibiotics were also elucidated using standard procedures. The aqueous and acetone extracts of the seed were screened for activity against 27 bacterial isolates. The aqueous extract exhibited activity mainly against Gram positive organisms with Minimum inhibitory concentration (MIC) values ranging from 5 mgml-1 – 20 mgml-1, while the acetone extract showed activity against both Gram negative and Gram positive organisms with MIC values ranging from 10 mgml-1 - 0.156 mgml-1. The acetone extract also showed rapid bactericidal activity against Staphylococcus aureus ATCC 6538 with a 3.097 Log10 reduction in counts within 4 hours at 0.3125 mgml-1 and a 1.582 Log10 reduction against Proteus vulgaris CSIR 0030 at 5 mgml-1 after 1 hour. In addition, the aqueous, methanol and acetone extracts of the seeds also exhibited activity against four clinical strains of Staphylococcus isolated from wound sepsis specimens. The MIC values for the aqueous extract were 10 mgml-1 for all the isolates while the acetone and methanol extracts had lower values ranging from 0.3125 - 0.625 mgml-1. The acetone extract was strongly bactericidal against Staphylococcus aureus OKOH3 resulting in a 2.70 Log10 reduction in counts at 1.25 mgml-1 within 4 hours of exposure and a complete elimination of the organism after 8 hours. The bactericidal vi activity of the same extract against Staphylococcus aureus OKOH1 was weak, achieving only a 2.92 Log10 reduction in counts at 1.25 mgml-1 (4× MIC) in 24 hours. In the test for interactions between the acetone extract of the seeds and antibiotics, synergistic interactions were observed largely against Gram positive organisms using the FIC indices, (indices of 0.52 - 0.875) with combinations against Gram negatives yielding largely antagonistic interactions (indices of 2.0 to 5.0). Synergy (≥ 1000 times or ≥ 3 Log10 potentiation of the bactericidal activity) against both Gram negative and Gram positive organisms was detected by time kill assays mainly involving the antibiotics tetracycline, chloramphenicol, amoxycillin and penicillin G. Combinations involving erythromycin and ciprofloxacin consistently gave antagonistic or indifferent interactions. We conclude that the acetone extract of Garcinia kola seeds possess strong bactericidal activities against both Gram positive and Gram negative organisms and can be therapeutically useful in the treatment of bacterial infections including the problematic staphylococcal wound infections. In addition, the acetone extract can be a potential source of broad spectrum resistance modifying compounds that can potentially improve the performance of antibiotics in the treatment of drug resistant infections.
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- Authors: Sibanda, Thulani
- Date: 2007
- Subjects: Drug resistance in microorganisms , Garcinia , Antibiotics , Medicinal plants
- Language: English
- Type: Thesis , Masters , MSc (Microbiology)
- Identifier: vital:11242 , http://hdl.handle.net/10353/71 , Drug resistance in microorganisms , Garcinia , Antibiotics , Medicinal plants
- Description: The antibacterial potency of the extracts of the seed of Garcinia kola (bitter kola) was investigated in this study against a panel of referenced, environmental and clinical bacterial strains. The killing rates of the active extract as well as their potential for combination antibacterial therapy with standard antibiotics were also elucidated using standard procedures. The aqueous and acetone extracts of the seed were screened for activity against 27 bacterial isolates. The aqueous extract exhibited activity mainly against Gram positive organisms with Minimum inhibitory concentration (MIC) values ranging from 5 mgml-1 – 20 mgml-1, while the acetone extract showed activity against both Gram negative and Gram positive organisms with MIC values ranging from 10 mgml-1 - 0.156 mgml-1. The acetone extract also showed rapid bactericidal activity against Staphylococcus aureus ATCC 6538 with a 3.097 Log10 reduction in counts within 4 hours at 0.3125 mgml-1 and a 1.582 Log10 reduction against Proteus vulgaris CSIR 0030 at 5 mgml-1 after 1 hour. In addition, the aqueous, methanol and acetone extracts of the seeds also exhibited activity against four clinical strains of Staphylococcus isolated from wound sepsis specimens. The MIC values for the aqueous extract were 10 mgml-1 for all the isolates while the acetone and methanol extracts had lower values ranging from 0.3125 - 0.625 mgml-1. The acetone extract was strongly bactericidal against Staphylococcus aureus OKOH3 resulting in a 2.70 Log10 reduction in counts at 1.25 mgml-1 within 4 hours of exposure and a complete elimination of the organism after 8 hours. The bactericidal vi activity of the same extract against Staphylococcus aureus OKOH1 was weak, achieving only a 2.92 Log10 reduction in counts at 1.25 mgml-1 (4× MIC) in 24 hours. In the test for interactions between the acetone extract of the seeds and antibiotics, synergistic interactions were observed largely against Gram positive organisms using the FIC indices, (indices of 0.52 - 0.875) with combinations against Gram negatives yielding largely antagonistic interactions (indices of 2.0 to 5.0). Synergy (≥ 1000 times or ≥ 3 Log10 potentiation of the bactericidal activity) against both Gram negative and Gram positive organisms was detected by time kill assays mainly involving the antibiotics tetracycline, chloramphenicol, amoxycillin and penicillin G. Combinations involving erythromycin and ciprofloxacin consistently gave antagonistic or indifferent interactions. We conclude that the acetone extract of Garcinia kola seeds possess strong bactericidal activities against both Gram positive and Gram negative organisms and can be therapeutically useful in the treatment of bacterial infections including the problematic staphylococcal wound infections. In addition, the acetone extract can be a potential source of broad spectrum resistance modifying compounds that can potentially improve the performance of antibiotics in the treatment of drug resistant infections.
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Design and implementation of a network revenue management architecture for marginalised communities
- Authors: Tarwireyi, Paul
- Date: 2007
- Subjects: Community development -- South Africa -- Eastern Cape , Computer network architectures -- South Africa -- Design and construction , Computer network protocols -- South Africa -- Eastern Cape -- Design and construction , Revenue management , Computer networks -- Social aspects
- Language: English
- Type: Thesis , Masters , MSc (Computer Science)
- Identifier: vital:11374 , http://hdl.handle.net/10353/72 , Community development -- South Africa -- Eastern Cape , Computer network architectures -- South Africa -- Design and construction , Computer network protocols -- South Africa -- Eastern Cape -- Design and construction , Revenue management , Computer networks -- Social aspects
- Description: Rural Internet connectivity projects aimed at bridging the digital divide have mushroomed across many developing countries. Most of the projects are deployed as community centred projects. In most of the cases the initial deployment of these projects is funded by governments, multilateral institutions and non-governmental organizations. After the initial deployment, financial sustainability remains one of the greatest challenges facing these projects. In the light of this, externally funded ICT4D interventions should just be used for “bootstrapping” purposes. The communities should be “groomed” to take care of and sustain these projects, eliminating as soon as possible a dependency on external funding. This master thesis presents the design and the implementation of a generic architecture for the management of the costs associated with running a computer network connected to the Internet, The proposed system, called the Network Revenue Management System, enables a network to generate revenue, by charging users for the utilization of network resources. The novelty of the system resides in its flexibility and adaptability, which allow the exploration of both conventional and non-conventional billing options, via the use of suitable ‘adapters’. The final goal of the exploration made possible by this system is the establishment of what is regarded as equitable charging in rural, marginalized communities - such as the community in Dwesa, South Africa.
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- Authors: Tarwireyi, Paul
- Date: 2007
- Subjects: Community development -- South Africa -- Eastern Cape , Computer network architectures -- South Africa -- Design and construction , Computer network protocols -- South Africa -- Eastern Cape -- Design and construction , Revenue management , Computer networks -- Social aspects
- Language: English
- Type: Thesis , Masters , MSc (Computer Science)
- Identifier: vital:11374 , http://hdl.handle.net/10353/72 , Community development -- South Africa -- Eastern Cape , Computer network architectures -- South Africa -- Design and construction , Computer network protocols -- South Africa -- Eastern Cape -- Design and construction , Revenue management , Computer networks -- Social aspects
- Description: Rural Internet connectivity projects aimed at bridging the digital divide have mushroomed across many developing countries. Most of the projects are deployed as community centred projects. In most of the cases the initial deployment of these projects is funded by governments, multilateral institutions and non-governmental organizations. After the initial deployment, financial sustainability remains one of the greatest challenges facing these projects. In the light of this, externally funded ICT4D interventions should just be used for “bootstrapping” purposes. The communities should be “groomed” to take care of and sustain these projects, eliminating as soon as possible a dependency on external funding. This master thesis presents the design and the implementation of a generic architecture for the management of the costs associated with running a computer network connected to the Internet, The proposed system, called the Network Revenue Management System, enables a network to generate revenue, by charging users for the utilization of network resources. The novelty of the system resides in its flexibility and adaptability, which allow the exploration of both conventional and non-conventional billing options, via the use of suitable ‘adapters’. The final goal of the exploration made possible by this system is the establishment of what is regarded as equitable charging in rural, marginalized communities - such as the community in Dwesa, South Africa.
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Development of a web-based interface for a wireless sensor network monitoring system
- Authors: Gumbo, Sibukele
- Date: 2007
- Subjects: Wireless LAN , Sensor networks , Wireless communication systems , Web sites -- Design , User interfaces (Computer systems)
- Language: English
- Type: Thesis , Masters , MSc (Computer Science)
- Identifier: vital:11372 , http://hdl.handle.net/10353/68 , Wireless LAN , Sensor networks , Wireless communication systems , Web sites -- Design , User interfaces (Computer systems)
- Description: In the recent past, wireless sensor technology has undergone advancements in its autonomous data collecting aspects, and has become an area worth investigating in relation to structural monitoring applications. The system described in this thesis aims at acquiring, storing and displaying overhead transmission line related data collected from a wireless sensor network. Open source tools were used in its development and implementation. The inherent linearly aligned topology of transmission line monitoring devices is not without shortcomings; hence analysis of linear node placement, hardware and software components was carried out to determine the feasibility of the system. Their limited data processing capabilities has motivated the development of a post processing wireless sensor application in order to present any collected structural data in an understandable format.
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- Authors: Gumbo, Sibukele
- Date: 2007
- Subjects: Wireless LAN , Sensor networks , Wireless communication systems , Web sites -- Design , User interfaces (Computer systems)
- Language: English
- Type: Thesis , Masters , MSc (Computer Science)
- Identifier: vital:11372 , http://hdl.handle.net/10353/68 , Wireless LAN , Sensor networks , Wireless communication systems , Web sites -- Design , User interfaces (Computer systems)
- Description: In the recent past, wireless sensor technology has undergone advancements in its autonomous data collecting aspects, and has become an area worth investigating in relation to structural monitoring applications. The system described in this thesis aims at acquiring, storing and displaying overhead transmission line related data collected from a wireless sensor network. Open source tools were used in its development and implementation. The inherent linearly aligned topology of transmission line monitoring devices is not without shortcomings; hence analysis of linear node placement, hardware and software components was carried out to determine the feasibility of the system. Their limited data processing capabilities has motivated the development of a post processing wireless sensor application in order to present any collected structural data in an understandable format.
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Rural internet connectivity: a development in Dwesa-Cwebe, Eastern Cape, South Africa
- Authors: Mandioma, Martin
- Date: 2007
- Subjects: Internet -- South Africa -- Eastern Cape
- Language: English
- Type: Thesis , Masters , MSc (Computer Science)
- Identifier: vital:11376 , http://hdl.handle.net/10353/78 , Internet -- South Africa -- Eastern Cape
- Description: This thesis presents aspects of Internet connectivity in rural South Africa. The work looks at government initiatives being undertaken to connect rural communities to up-to-date information networks. Various projects that seek to connect rural areas of South Africa, as well as other remote areas around the world, are discussed. These projects present many novel ideas that have been successfully used to link rural communities in remote areas with the information age. In particular, wired and wireless access technologies that can be implemented to connect remote communities to the Internet are discussed. A field test utilizing GPRS, VSAT and WiMAX was implemented in Dwesa-Cwebe, Eastern Cape Province, South Africa. VSAT proved to offer better Internet connectivity in terms of throughput and latency. WiMAX was then successfully implemented to relay the signal over the remote area of Dwesa-Cwebe, thus effectively providing Internet connectivity to an area with limited cell phone coverage and no telephone lines.
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- Authors: Mandioma, Martin
- Date: 2007
- Subjects: Internet -- South Africa -- Eastern Cape
- Language: English
- Type: Thesis , Masters , MSc (Computer Science)
- Identifier: vital:11376 , http://hdl.handle.net/10353/78 , Internet -- South Africa -- Eastern Cape
- Description: This thesis presents aspects of Internet connectivity in rural South Africa. The work looks at government initiatives being undertaken to connect rural communities to up-to-date information networks. Various projects that seek to connect rural areas of South Africa, as well as other remote areas around the world, are discussed. These projects present many novel ideas that have been successfully used to link rural communities in remote areas with the information age. In particular, wired and wireless access technologies that can be implemented to connect remote communities to the Internet are discussed. A field test utilizing GPRS, VSAT and WiMAX was implemented in Dwesa-Cwebe, Eastern Cape Province, South Africa. VSAT proved to offer better Internet connectivity in terms of throughput and latency. WiMAX was then successfully implemented to relay the signal over the remote area of Dwesa-Cwebe, thus effectively providing Internet connectivity to an area with limited cell phone coverage and no telephone lines.
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The classification of some fuzzy subgroups of finite groups under a natural equivalence and its extension, with particular emphasis on the number of equivalence classes
- Authors: Ndiweni, Odilo
- Date: 2007
- Subjects: Fuzzy sets , Maximal functions , Finite groups , Equivalence classes (Set theory)
- Language: English
- Type: Thesis , Masters , M Sc (Mathematics)
- Identifier: vital:11587 , http://hdl.handle.net/10353/88 , Fuzzy sets , Maximal functions , Finite groups , Equivalence classes (Set theory)
- Description: In this thesis we use the natural equivalence of fuzzy subgroups studied by Murali and Makamba [25] to characterize fuzzy subgroups of some finite groups. We focus on the determination of the number of equivalence classes of fuzzy subgroups of some selected finite groups using this equivalence relation and its extension. Firstly we give a brief discussion on the theory of fuzzy sets and fuzzy subgroups. We prove a few properties of fuzzy sets and fuzzy subgroups. We then introduce the selected groups namely the symmetric group 3 S , dihedral group 4 D , the quaternion group Q8 , cyclic p-group pn G = Z/ , pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We also present their subgroups structures and construct lattice diagrams of subgroups in order to study their maximal chains. We compute the number of maximal chains and give a brief explanation on how the maximal chains are used in the determination of the number of equivalence classes of fuzzy subgroups. In determining the number of equivalence classes of fuzzy subgroups of a group, we first list down all the maximal chains of the group. Secondly we pick any maximal chain and compute the number of distinct fuzzy subgroups represented by that maximal chain, expressing each fuzzy subgroup in the form of a keychain. Thereafter we pick the next maximal chain and count the number of equivalence classes of fuzzy subgroups not counted in the first chain. We proceed inductively until all the maximal chains have been exhausted. The total number of fuzzy subgroups obtained in all the maximal chains represents the number of equivalence classes of fuzzy subgroups for the entire group, (see sections 3.2.1, 3.2.2, 3.2.6, 3.2.8, 3.2.9, 3.2.15, 3.16 and 3.17 for the case of selected finite groups). We study, establish and prove the formulae for the number of maximal chains for the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . To accomplish this, we use lattice diagrams of subgroups of these groups to identify the maximal chains. For instance, the group pn qm G = Z/ + Z/ would require the use of a 2- dimensional rectangular diagram (see section 3.2.18 and 5.3.5), while for the group pn qm r s G = Z/ + Z/ + Z/ we execute 3- dimensional lattice diagrams of subgroups (see section 5.4.2, 5.4.3, 5.4.4, 5.4.5 and 5.4.6). It is through these lattice diagrams that we identify routes through which to carry out the extensions. Since fuzzy subgroups represented by maximal chains are viewed as keychains, we give a brief discussion on the notion of keychains, pins and their extensions. We present propositions and proofs on why this counting technique is justifiable. We derive and prove formulae for the number of equivalence classes of the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We give a detailed explanation and illustrations on how this keychain extension principle works in Chapter Five. We conclude by giving specific illustrations on how we compute the number of equivalence classes of a fuzzy subgroup for the group p2 q2 r 2 G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of the group p q r G = Z/ + Z/ + Z/ 1 2 2 . This illustrates a general technique of computing the number of fuzzy subgroups of G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of 1 -1 = / + / + / pn qm r s G Z Z Z . Our illustration also shows two ways of extending from a lattice diagram of 1 G to that of G .
- Full Text:
- Authors: Ndiweni, Odilo
- Date: 2007
- Subjects: Fuzzy sets , Maximal functions , Finite groups , Equivalence classes (Set theory)
- Language: English
- Type: Thesis , Masters , M Sc (Mathematics)
- Identifier: vital:11587 , http://hdl.handle.net/10353/88 , Fuzzy sets , Maximal functions , Finite groups , Equivalence classes (Set theory)
- Description: In this thesis we use the natural equivalence of fuzzy subgroups studied by Murali and Makamba [25] to characterize fuzzy subgroups of some finite groups. We focus on the determination of the number of equivalence classes of fuzzy subgroups of some selected finite groups using this equivalence relation and its extension. Firstly we give a brief discussion on the theory of fuzzy sets and fuzzy subgroups. We prove a few properties of fuzzy sets and fuzzy subgroups. We then introduce the selected groups namely the symmetric group 3 S , dihedral group 4 D , the quaternion group Q8 , cyclic p-group pn G = Z/ , pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We also present their subgroups structures and construct lattice diagrams of subgroups in order to study their maximal chains. We compute the number of maximal chains and give a brief explanation on how the maximal chains are used in the determination of the number of equivalence classes of fuzzy subgroups. In determining the number of equivalence classes of fuzzy subgroups of a group, we first list down all the maximal chains of the group. Secondly we pick any maximal chain and compute the number of distinct fuzzy subgroups represented by that maximal chain, expressing each fuzzy subgroup in the form of a keychain. Thereafter we pick the next maximal chain and count the number of equivalence classes of fuzzy subgroups not counted in the first chain. We proceed inductively until all the maximal chains have been exhausted. The total number of fuzzy subgroups obtained in all the maximal chains represents the number of equivalence classes of fuzzy subgroups for the entire group, (see sections 3.2.1, 3.2.2, 3.2.6, 3.2.8, 3.2.9, 3.2.15, 3.16 and 3.17 for the case of selected finite groups). We study, establish and prove the formulae for the number of maximal chains for the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . To accomplish this, we use lattice diagrams of subgroups of these groups to identify the maximal chains. For instance, the group pn qm G = Z/ + Z/ would require the use of a 2- dimensional rectangular diagram (see section 3.2.18 and 5.3.5), while for the group pn qm r s G = Z/ + Z/ + Z/ we execute 3- dimensional lattice diagrams of subgroups (see section 5.4.2, 5.4.3, 5.4.4, 5.4.5 and 5.4.6). It is through these lattice diagrams that we identify routes through which to carry out the extensions. Since fuzzy subgroups represented by maximal chains are viewed as keychains, we give a brief discussion on the notion of keychains, pins and their extensions. We present propositions and proofs on why this counting technique is justifiable. We derive and prove formulae for the number of equivalence classes of the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We give a detailed explanation and illustrations on how this keychain extension principle works in Chapter Five. We conclude by giving specific illustrations on how we compute the number of equivalence classes of a fuzzy subgroup for the group p2 q2 r 2 G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of the group p q r G = Z/ + Z/ + Z/ 1 2 2 . This illustrates a general technique of computing the number of fuzzy subgroups of G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of 1 -1 = / + / + / pn qm r s G Z Z Z . Our illustration also shows two ways of extending from a lattice diagram of 1 G to that of G .
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