Symmetry properties for first integrals

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. July 2014.This is the study of Lie algebraic properties of first integrals of scalar second-, third and higher-order ordinary differential equations (ODEs). The Lie algebraic classification of such differential equations is now well-known from the works of Lie [10] as well as recently Mahomed and Leach [19]. However, the algebraic properties of first integrals are not known except in the maximal cases for the basic first integrals and some of their quotients. Here our intention is to investigate the complete problem for scalar second-order and maximal symmetry classes of higher-order ODEs using Lie algebras and Lie symmetry methods. We invoke the realizations of low-dimensional Lie algebras. Symmetries of the fundamental first integrals for scalar second-order ODEs which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the point symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classi cation of point symmetries of first integrals of such linear ODEs is studied. As a consequence, we provide a counting theorem for the point symmetries of first integrals of scalar linearizable second-order ODEs. We show that there exists the 0, 1, 2 or 3 point symmetry cases. It is proved that the maximal algebra case is unique. By use of Lie symmetry group methods we further analyze the relationship between the first integrals of the simplest linear third-order ODEs and their point symmetries. It is well-known that there are three classes of linear third-order ODEs for maximal and submaximal cases of point symmetries which are 4, 5 and 7. The simplest scalar linear third-order equation has seven point symmetries. We obtain the classifying relation between the symmetry and the first integral for the simplest equation. It is shown that the maximal Lie algebra of a first integral for the simplest equation y000 = 0 is unique and four-dimensional. Moreover, we show that the Lie algebra of the simplest linear third-order equation is generated by the symmetries of the two basic integrals. We also obtain counting theorems of the symmetry properties of the first integrals for such linear third-order ODEs of maximal type. Furthermore, we provide insights into the manner in which one can generate the full Lie algebra of higher-order ODEs of maximal symmetry from two of their basic integrals. The relationship between rst integrals of sub-maximal linearizable third-order ODEs and their symmetries are investigated as well. All scalar linearizable third-order equations can be reduced to three classes by point transformations. We obtain the classifying relations between the symmetries and the first integral for sub-maximal cases of linear third-order ODEs. It is known, from the above, that the maximum Lie algebra of the first integral is achieved for the simplest equation. We show that for the other two classes they are not unique. We also obtain counting theorems of the symmetry properties of the rst integrals for these classes of linear third-order ODEs. For the 5 symmetry class of linear third-order ODEs, the first integrals can have 0, 1, 2 and 3 symmetries and for the 4 symmetry class of linear third-order ODEs they are 0, 1 and 2 symmetries respectively. In the case of sub-maximal linear higher-order ODEs, we show that their full Lie algebras can be generated by the subalgebras of certain basic integrals. For the n+2 symmetry class, the symmetries of the rst integral I2 and a two-dimensional subalgebra of I1 generate the symmetry algebra and for the n + 1 symmetry class, the full algebra is generated by the symmetries of I1 and a two-dimensional subalgebra of the quotient I3=I2. Finally, we completely classify the first integrals of scalar nonlinear second-order ODEs in terms of their Lie point symmetries. This is performed by first obtaining the classifying relations between point symmetries and first integrals of scalar nonlinear second order equations which admit 1, 2 and 3 point symmetries. We show that the maximum number of symmetries admitted by any first integral of a scalar second-order nonlinear (which is not linearizable by point transformation) ODE is one which in turn provides reduction to quadratures of the underlying dynamical equation. We provide physical examples of the generalized Emden-Fowler, Lane-Emden and modi ed Emden equations

A Corpus-Based Study of Additive and Adversative in Clause Complex: The case of British and Pakistani Research Discourse

The present study aims at investigating the frequency distributions of conjunctive relations such as additive and adversative in native and non-native Pakistani research discourse by applying the frame work of conjunctive relations(Halliday & Hasan, 1976).In order to achieve the objectives of the present study, two corpora native and non-native are developed each comprising of one million words. The analysis of native and non-native research discourse is carried out by utilizing the mixed method approach (QUAN→qual) (see Creswell ,2007).The study reveals that additives are more frequently(9.36%) used in the Pakistanicorpus as compared to that of native corpus. On the other hand, adversative conjunctions are more frequently(6.32%)used in the native corpus as compared to that of non-native Pakistani corpus. The findings expose that non-native corpus showed 1.51% lower variety of conjunctive relations as compared to that of native corpus.The study implies that non-native researcher’s access to the native research discourse will enable them to widen their knowledge about the correct and multiple use of additive and adversative conjunctive relations in their research discourse

محمد عاصم بٹ کے ناولوں میں وجودیت کے اثرات

Existentialism is the intellectual marquee of the present age. The specific anxieties, tensions, anxieties of this age, which are the result of horrific wars, destructive weapons, violent social revolutions, authoritarian, sectarian and anti-individual beliefs and the decline of spirituality.  It has happened. It has caused confusion and despair, rebellion against all accepted traditions, materialism, depravity of imagination, disintegration of spirit and unity in thought, literature and culture, feeling of insecurity, social, political, religious, The defeat of moral and aesthetic values, from which our culture has been shaped, has provided the raw material for the philosophy of existence, which is its interpretation, its interpretation, its derivation, and its critique.  Man was the need of proof of his own caste, and he himself was rich in the caste of self-belief, determination, courage and courage, was the embers of centuries and the lone man. But when does man lose courage. Moments full of anguish and sorrow and grief.  Human life and society have been the focus of attention in Urdu novels from the very beginning. That is why Asim Butt, who is a modern novelist, covered modern philosophical and intellectual discussions. Asim Butt  Kept an eye on the changing circumstances and made the problems of human existence his subject. Muhammad Asim butt is one of the most prominent novelist of 21 century. He wrote three novels Daira, Natamam, Bhaeed. In this article his three novels in   Existential effects has been  discussed.</p

Identification of Challenges for Social Inclusion of Illiterate Deaf Adults

Purpose: This qualitative case study investigates the profound challenges hindering the social inclusion of illiterate deaf adults within Gujranwala city. Methodology: A sample of eight illiterate deaf adults and sixteen family members was meticulously selected through snowball sampling. Data collection utilized a tailored semi-structured interview schedule, validated by experts, and conducted in sign language. Findings: Thematic analysis revealed a multitude of challenges, including communication barriers, unfamiliarity with standardized sign language, low self-confidence, limited awareness of civil and legal rights, attitudinal barriers, shyness, and hesitance. These challenges collectively undermine the social inclusion of illiterate deaf adults. Implications: The study underscores the pressing need for adult education initiatives tailored to their unique needs to facilitate their independent integration into society

Educational Experiences of Students with Hearing Impairment Studying in Special Education Institutions across Punjab

Purpose: This qualitative research looks into the nuanced educational experiences of students with hearing impairment within special education institutions across the nine divisions of Punjab. Design/Methodology/Approach: This research employed a qualitative study design with a focus on thematic analysis. The study population consisted of students with hearing impairment, and a purposive sample of 36 participants was selected for in-depth interviews. Findings: Through in-depth interviews with 36 participants, we unearthed prevailing circumstances revealing a landscape characterized by traditional teaching approaches, limited use of technology, and disparities between special education centers and schools. Elementary-level students encounter greater challenges, as secondary-level students prepare for board exams, driving heightened teacher scrutiny. Likewise students studying in special education centres face greater challenges than in schools. Implications/Originality/Value: These findings underscore the need for a more inclusive and technologically enriched educational framework, accompanied by personalized support services. By addressing these disparities and providing tailored education, we can empower students with hearing impairment to thrive academically and holistically

Analyzing the Implementation of the Special Education Program for Students with Hearing Impairment at Primary School Level in Punjab

The purpose of the study was to analyze the implementation of the special education program for students with hearing impairment (SWHI) at primary school level in Punjab. The nature of the research was descriptive. Teachers of SWHI working at special education primary schools were selected as population. Multi-stage random sampling technique was used to select the sample which was comprised of 209 teachers of SWHI (M=43 &amp; F=166). Data was collected by survey method through self-developed and validated questionnaire having reliability of α= 0.92. Independent sample t-test and one-way ANOVA were run to draw the results of the study. Findings of the study indicated that there are very few constructive activities are being practiced such as daily lesson planning, assigning homework, taking weekly &amp; monthly test, activity based instructions, formative assessment, providing immediate feedback on performance. But many contributing practices were found to be employed in a passive manner like knowledgeable classroom management and instructional strategies, training to deal assistive technological devices, parent-teacher collaboration, teachers’ education program, and use of audio-visual aids. Workload, extra duties, lack of technological resources, use of local sign language, neglected teacher-professional collaboration, higher student-teacher ratio, dealing more than one class at same time were the flaws affecting the quality of process of primary special education program in Punjab. It was recommended that the role of related stakeholders such as government, head teachers, teachers, non-teaching staff, parents and community should be reaffirmed in true sense

Challenges Limiting the Role of Deaf Parents in Academics of their Children with Normal Hearing

Purpose: The goal of this research was to investigate the difficulties that deaf parents have when it comes to the academic performance of their hearing children. Methodology: Case studies based on qualitative paradigms were used for this study. Participants in the research were parents who were deaf themselves but whose children had normal hearing. The sample comprises of 14 deaf couples from two divisions of Punjab—Lahore and Gujranwala. The data collection method consisted of a self-developed interview schedule with open-ended questions. Thematic analysis, a qualitative method, was used to analyze the data. Findings: Deaf parents have to deal with a number of obstacles, the most significant of which are communication barriers, attitudes they confront, and misunderstandings held by the community as a whole, all of which limit their capacity to participate in their children's academic lives who have normal hearing. Implications: It was suggested to the various stakeholders that they should be required to play their respective contributing roles in the process of reducing the constraints that restrict the engagement of parents in the academics of their children on a consistent basis, beginning with the most fundamental level.                                                                                                        

Challenges Limiting the Role of Deaf Parents in Academics of their Children with Normal Hearing

Purpose: The goal of this research was to investigate the difficulties that deaf parents have when it comes to the academic performance of their hearing children. Methodology: Case studies based on qualitative paradigms were used for this study. Participants in the research were parents who were deaf themselves but whose children had normal hearing. The sample comprises of 14 deaf couples from two divisions of Punjab—Lahore and Gujranwala. The data collection method consisted of a self-developed interview schedule with open-ended questions. Thematic analysis, a qualitative method, was used to analyze the data. Findings: Deaf parents have to deal with a number of obstacles, the most significant of which are communication barriers, attitudes they confront, and misunderstandings held by the community as a whole, all of which limit their capacity to participate in their children's academic lives who have normal hearing. Implications: It was suggested to the various stakeholders that they should be required to play their respective contributing roles in the process of reducing the constraints that restrict the engagement of parents in the academics of their children on a consistent basis, beginning with the most fundamental level