Smoothness conditions and symmetries of partial differential equations

A research report submitted to the Faculty of Science, University of the Witwatersrand, in ful lment of the requirements for the degree of Master of Science. School of Mathematics Johannesburg February 15, 2016We obtain a solution of the Black-Scholes equation with a non-smooth bound- ary condition using symmetry methods. The Black-Scholes equation along with its boundary condition are rst transformed into the one dimensional heat equation and an initial condition respectively. We then nd an appro- priate general symmetry generator of the heat equation using symmetries of the heat equation and the fundamental solution of the heat equation. The method we use to nd the symmetry generator is such that the boundary condition is left invariant and yet the symmetry can still be used to solve the heat equation. We then use the help of Mathematica to nd the solution to the heat equation. Then the solution is then transformed backwards to a solution of the Black-Scholes equation using the same change of variables that were used for the forward transformations. The solution is then nally checked if it satis es the boundary condition of the Black-Scholes equation

READING ACROSS BOUNDARIES: CULTURAL CONFLICTS AND REMEDIES

Reading, according to Butterflies et al. (2020), is a cognitive process involving the decoding of written symbols to extract or construct meaning. However, from a pragmatist perspective, this notion is outdated and insufficient. Reading is no longer merely a passive act of decoding thoughts. Instead, it should be perceived as an active and dynamic interaction between the reader and the text. A student interpretation of a text is significantly influenced by their schemata, which includes cultural background knowledge, personal experiences, and prior understanding. This perspective does not discount the crucial role of linguistic knowledge in reading comprehension. Instead, it highlights the fact that cultural knowledge is equally essential for achieving a complete and genuine understanding of the text. Linguistic and cultural knowledge are interconnected and should not be viewed in isolation. When these elements are separated, the reader fails to grasp the full significance and richness of what they are reading, resulting in an incomplete and superficial interpretation. This paper argues  that to fully comprehend a text, both linguistic and cultural contexts must be considered together, providing a holistic and meaningful reading experience for students.  Article visualizations

Rare earth doped Titania/Carbon nanomaterials composite photocatalysts for water treatment

PhD. (Chemistry)Pre-synthesised gadolinium oxide decorated multiwalled carbon nanotubes (MWCNT-Gd) were coupled with titania to form nanocomposite photocatalysts (MWCNT-Gd/TiO2) using a sol-gel method. Rare earth metal ions (Eu, Nd and Gd), nitrogen and sulphur tridoped titania were decorated on MWCNT-Gd to yield composite photocatalysts (MWCNT-Gd/Eu/Nd/Gd/N,S-TiO2) by a similar method, using thiourea as nitrogen and sulphur source. Different carbon nanomaterials were incorporated into tridoped titania to form various composite photocatalysts (MWCNT/Gd,N,S-TiO2, MWCNT/Nd,N,S-TiO2, SWCNT (single walled carbon nanotube)/Nd,N,S-TiO2 and rGO (reduced graphene oxide)/Nd,N,S-TiO2) via the sol-gel method. Likewise, gadolinium doped graphitic carbon nitride (g-C3N4-Gd3+) was obtained by heating a mixture of gadolinium nitrate hexahydrate and cyanoguanidine and subsequently hybridised with MWCNT/TiO2 using the sol-gel method to yield composite photocatalysts with varying g-C3N4-Gd3+ loadings. All the prepared photocatalysts were characterised by microscopic tools (FE/FIB-SEM-EDX, TEM), crystallographic technique (XRD), spectroscopic tools (UV-Vis, Raman and FT-IR) and nitrogen sorption technique (BET)

Spectroscopic determination of water salinity in brackish surface water in Nandoni Dam, at Vhembe District, Limpopo Province, South Africa

The problem of limited water supply in the Vhembe District (Limpopo Province, South Africa) is exacerbated by a preponderance of dissolved salts, which cause disagreeable taste and odour in the water as reported by the communities using this water for drinking. The water treatment plant that supplies the treated water to the communities in the District sources this raw water from the Nandoni Dam at the Luvuvhu river catchment. There are no scientific studies that have been reported in the literature that focused on determining the levels of water salinity from various water sources in the municipalities of the District. Water samples from various sites across the Nandoni Dam, a primary source of domestic water supply in the region, were collected through each season over a period of twelve months in order to ascertain the concentrations of dissolved salts in the dam. Onsite analyses of the water samples were conducted using the YSI ProDSS multimeter, while the laboratory water analyses were conducted using the spectroquant and atomic absorption spectrometers. Although salinity tests seem to indicate that the water sampled across most of the Nandoni Dam is brackish during all seasons of the year with the highest being 750 mg/L, water samples from the dam mid-outlet and the treatment plant are slightly below the World Health Organization (WHO) brackish water bracket of 500 mg/L with unfavourable taste for drinking. Results from this study indicate that the water sourced from the Nandoni Dam is not suitable for human consumption and therefore requires integrated water resource management, as well as robust and cost-effective water desalination treatment

Learning Styles and College Readiness of Philippine K-12 Graduates

Learning styles are crucial in determining students' increased academic achievement. Nevertheless, despite the vast amount of empirical research that has examined how learning styles relate to academic preparation and success, studies still need to be undertaken on how learning styles are linked to the preparedness of Philippine K-12 graduates for college life. Thus, the study determined the K-12 graduates' learning styles, examined their college readiness, and ascertained the significant difference in their college readiness when grouped according to their learning styles. Using descriptive-comparative design and 7,384 first-year students (2925 males, 4459 females) enrolled at a public Philippine university for the Academic Year (AY) 2019–2020, it was found that the Philippine K-12 graduates have a neutral learning style, which reflects their lack of a specific or focused learning style responsive to developing certain learning competencies. Also, the K-12 graduates were collegeunready as they fell short of demonstrating acceptable competencies defined by the College Readiness Standards (CRS). Remarkably, the study has established that college readiness significantly varies in terms of learning styles. In particular, the K-12 graduates who are social-conceptual learners are more college ready than those who exhibit independent, neutral, and social learning styles. Lastly, implications to basic and higher education instructional practices were presented based on the study findings

AUTHENTIC MATERIALS: FOSTERING COLLABORATIVE LANGUAGE LEARNING

Authentic materials serve as a framework in language education, placing emphasis on resources that resonate with the everyday experiences of learners. This approach ignites a profound motivation among learners. They are prompting their active involvement in the learning process. By infusing authentic materials into the lesson, students get exposed to a practical and relatable learning experience that, as they provide real-world contexts, fosters a more enhanced comprehension and better retention. The incorporation of authentic materials represents more than just a pedagogy; it symbolizes an initial venture into the vigorous intricacies of language acquisition, where learners are invited to embark on a learning journey filled with excitement and discovery. Confronting linguistic challenges akin to deciphering a foreign tongue can indeed prove daunting. Nevertheless, the burden of learning is significantly alleviated when materials resonate with students on a personal level, offering relatable content that resonates with their interests and experiences. Moreover, the art of tailoring materials to meet the unique needs and backgrounds of students is a transformative practice that yields tangible results in their academic journey. By aligning educational resources with students' cognitive and cultural landscapes, educators can cultivate an inclusive learning environment where every learner feels valued and empowered to excel. Ultimately, the integration of authentic materials not only enriches the educational experience but also fosters a sense of fulfilment and accomplishment among both students and educators alike, thereby propelling the trajectory of language learning to new heights of excellence.  Article visualizations

An assessment of occupational health and safety in the informal car maintenance,welding and spraypainting industry in Mbabane.

Student Number : 0215978H - MPh research report - School of Public Health - Faculty of Health SciencesThe study was conducted in the City of Mbabane and assessed Occupational Health and Safety in the informal car maintenance, welding and spray-painting industry. The objective of the study was to assess the risks workers are exposed to in the informal car maintenance, welding and spray-painting industry in Mbabane city in Swaziland. Data was collected by administration of a questionnaire to managers of the establishments and by personal observations of workers while on the job from walk through surveys that were conducted in the city. Seventy (70) workplaces were identified and sixty five (65) of them participated in the study, representing a response rate of 92.86%. There were twenty (20) workplaces doing car maintenance, twenty five (25) doing welding and twenty (20) doing spray painting. Fifty-three (53) The data was analysed using the EPI INFO software and results revealed that most workers in this sector were indeed at high risk of exposure to occupational health and safety problems. They worked under unfavourable conditions such as working in the open and subjected to adverse weather conditions, exposed to solvents, welding fumes and gases, strenuous work, improper postures, lifting heavy loads, exposed to spray painting aerosols and fumes and exposed to dust. The workers’ occupational health and safety was made worse by the fact that most of them did not have or use personal protective equipment. 90% of the workers were exposed to emissions while carrying out their jobs of spray painting and 10% of them were exposed to paint. All the workers that were doing spray painting were exposed to paint (95%) and solvents 5%). Although 75% of the workers, doing spray painting had some kind of personal protection provided however the usage rate was very low. In all the workplaces that were doing spray painting, there were no other existing control measures for protecting the workers from paint emissions 76% did not have any respiratory protection. However, only 33.3% of them were using the PPE provided and 66.7% were not using them. Therefore most of the workers were at risk of breathing in welding fumes and other welding related gases. This means 92 % of workers were at risk to welding fumes and gases. 68% of the workers did not have protection for the hands, only 32% had. Those workers that had hand protection (32%) had gloves with shorter cuffs and separate sleeves (12%). Others had leather gauntlet gloves with canvas or cuffs (20%). 75% of these workers who had PPE were not using them, only 25% did. Since most of the workers did not use hand protection, this means that their hands were not protected against heat, spatter, and radiation. Most of the workers (72%) did not wear eye protection when removing slag and that put them at risk of eye injuries. All welding operations were not done in a booth. This means that the workers and co-workers were at risk of exposure to welding gases and fumes. 48% of the workplaces had their surroundings with materials that could catch fire. 52% had their surroundings free from burnable material. Therefore almost half of the workplaces were at risk of catching fire. 76% of the working places had no fire extinguishers. Only 24% had fire extinguishers, but only two had been serviced accordingly. The workplaces were less prepared for outbreaks of fire. 68% of the workers took no precautions against burns; they had their sleeves rolled up and forearms without gloves or sleeves when carrying out their work. Only 32% of the workers took precautions against burns. 72% of the workers said that they did not know how to treat burns. Only 28% said they knew how to treat them. 72% of the workplaces did not have first aid kits. The means that they were not prepared for accident, only 28% had first aid kits. 71.4% of the work places had first aid kits without the necessary medicines, bandages, and equipment, only 28.6% had. This indicated a lack of preparedness for accidents on their part. A long-term strategy should be developed aimed at improving the occupational health and safety of the workers. Workers need to be empowered to perform their tasks safely. Workers and owners of informal industries should participate in the formulation of interventions aimed at improving occupational health and safety. The City Council should provide health and safety education and training to the Informal Sector

Prevention within a pastoral strategy : assessing the ABC-model with reference to the HIV/AIDS pandemic in Swaziland

Thesis (MTh)--Stellenbosch University, 2012.ENGLISH ABSTRACT: This thesis critically analyses the appropriateness and effectiveness of the ABC model in the HIV/AIDS prevention within the context of Swaziland. According to Louw (2008:423), the ABC model stands for the following: A = Abstinence, B = Be faithful and C = use Condoms. According to Green and Herling (2007:1) the ABC model has gained the attention of many countries. The attempt of this model in HIV prevention was to “aim at empowering people through value-based programs to basically abstain from sex as long as possible, to be faithful to one intimate partner and to use condoms correctly and consistently” (De la Porte 2006:79). The assessment of the HIV/AIDS virus in 1983-1984 came as a shock. It was difficult for the church to know at that time how to respond. Some of the responses pointed to the virus as punishment of God. Gradually the church started to become involved in the pandemic. From a Christian spirituality perspective it was argued that the so called ABC model could be viewed as a means of remedy within a prevention approach. The cultural issues as well as human sexuality factors have been discovered to be an obstacle in the ABC model in playing a progressive role in Swaziland. The cultural factors that prevent the ABC model from accomplishing effective results are listed in the final report of (Whiteside et al. 2006: 18-19): bunganwa (having multiple sexual partners; a cultural practice of male married and unmarried to have many girlfriends); kungenwa (levirate or wife inheritance); a widow is given in marriage to marry the brother of her deceased husband without the consent of the women which exposes women to the HIV virus. This practice is done without the consent of the women. Kujuma (occasional overnight visits between unmarried lovers); kuhlanta (a young girl bearing the children of her infertile sister); kushenda (extramarital relationships); kulamuta (a man having a sexual relationship with a younger sister of his wife); and sitsembu (polygamy, one man with more than one wife); this is a common cultural practice found in many African countries due to gender inequality (Chitando 2009:26). This has led to Swaziland being seriously affected by the HIV/AIDS virus (Rupiya 2006:66). The high rate of infection is frightening in a small country with a population of 1 million. As Bishop M, Mabuza, the Anglican bishop indicated, the nation‟s existence is threatened. (Rosenow 2011: 32). Therefore the researcher proposed a pastoral model which gives dignity to human and adds the spiritual dimension of healing in the pastoral care and counselling. There is an urgent need for church leaders to be empowered to face the HIV/AIDS with knowledge that the pandemic is not a punishment for promiscuous people. The theology of sexuality emphasises the responsibility in every sexual engagement for people to be conscious to whatever decision they take.AFRIKAANSE OPSOMMING: In hierdie navorsing word die toepaslikheid en effektiwiteit van die ABC model in die voorkoming van MIV/Vigs binne die konteks van Swaziland krities ontleed. Volgens Louw (2008:423) staan ABC (in Engels) vir: A = Onthouding, B = Getrouheid aan een maat en C = Die gebruik van kondome. Die navorser, ‟n geordende predikant van die Evangeliese Lutherse Kerk, bespreek die erns van die voorkoms van die MIV/Vigs-infeksie in Swaziland. Empiries is bewys dat Swaziland een van die lande in Sub-Sahara en die wêreld is met die hoogste infeksiekoers (Rupiya 2006:66). Dit is ernstig in ‟n klein landjie soos Swaziland met ‟n bevolking van slegs 1 129 000. Die pandemie het ‟n impak op alle gemeenskappe in die land en bied‟n groot uitdaging vir die Kerk in Swaziland. Die vraag is hoe die Kerk, in haar pastorale bediening, die ABC model kan gebruik om die globale poging te steun om te verhoed dat MIV/Vigs versprei. Hoe kan die konsep van die ABC model toegepas word sodat die Kerk, as hulpgewende gemeenskap, kan sorg vir die siele van die gemeentelede (cura animarum) en effektief inligting omtrent die ABC model kan versprei en uitreik na MIV-positiewe mense? Die navorser stel voor dat daar verandering moet kom in die gesindheid van die Kerk en dat daar ‟n herkonseptualisasie van die voorkomingstrategie en die Skrifverklaring van menslike seksualiteit moet wees. Die Kerke in Swaziland moet hulle houding verander. Daar moet baie meer openlikheid wees sodat sosio-kulturele kwessies openlik bespreek kan word. Daar moet ‟n kritiese herbepaling wees van die tradisies en die kerkbeleid rondom menslike seksualiteit en hoe dit betrekking het op die MIV/Vigs-pandemie

Challenges Faced by Learners with Partial Hearing Impairment in Lubombo Mainstream High Schools: A Wellness Perspective

This study sought to explore the challenges faced by learners with partial hearing impairment (LWPHI) in Lubombo mainstream high schools. The study used a social constructivism research paradigm and a qualitative research method. A phenomenological research design was utilised to find out how mainstream high school environment can handicap the LWPHI from reaching the vital and most cherished goal of independent living. Participants in the study were selected using purposive criterion sampling and they were 14 (n=14) in total. The data collection process entailed utilisation of individual semi-structured interviews and conducting of non-participant observation. Data was analysed using thematic analysis. The findings suggest that restrictive school environment robbed LWPHI equalized opportunities for meaningful participation in educational activities. Hence, modified instructional strategies and social environment are pertinent if the education sector really needs to combat academic discrepancies between the ‘normal’ and the LWPHI. Keywords: hearing impairment, inclusive education, learners, mainstream schools. DOI: 10.7176/JEP/10-27-05 Publication date:September 30th 201