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Müntz Space Embeddings of Schatten-Von Neumann Class

Thesis Info

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Author

Noor, Sahibzada Waleed

Supervisor

Dan Timotin

Program

PhD

Institute

Government College University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2007

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/2050/1/1710S.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726737000

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This thesis contains results about the embeddings of M ̈ untz spaces in the Hilbert space scenario and its applications to composition operators on M ̈ untz spaces. In the main, we shall be concerned with the embedding M Λ 2 ⊂ L 2 (μ), where the Hilbert- M ̈ untz space M Λ 2 is the closed linear span of the monomials x λ n in L 2 ([0, 1]) and μ is a finite Borel measure on [0, 1]. After gathering together the mathematical preliminaries required for this work in Chapter 1, we shall use the notion of a sublinear measure introduced by I. Chal- endar, E. Fricain and D. Timotin [8] to investigate the properties of boundedness, compactness and belonging to Schatten-von Neumann ideals of these Hilbert space embeddings. This will be the content of Chapters 2 and 3. In Chapter 4, we give ex- amples of sublinear measures for bounded and compact embeddings with interesting properties. Finally, in Chapter 5 the general embedding theory is applied to initiate the study of composition operators on M Λ 2 .
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مولانا محمد عثمان

مولانا محمد عثمان
افسوس ہے چند ماہ ہوئے مولانا محمد عثمان صاحب کا بھی اپنے وطن مالیگاؤں میں انتقال ہوگیا۔مولانا دارالعلوم دیوبند کے فارغ التحصیل تھے، استعداد پختہ تھی۔ یوں تو ان کومناسبت اوردلچسپی کم وبیش ہرفن سے تھی تاہم تفسیر اور حدیث ان کا خاص فن تھے۔ ایک عرصہ تک مالیگاؤں کے مختلف مدارس میں استاد رہے لیکن ان کی زندگی کاسب سے بڑااوراہم کارنامہ جوان کے لیے بقائے دوام کا ضامن ہے، مسلمان لڑکیوں کے لیے درس نظامی کاوہ عظیم الشان مدرسہ ہے جومالیگاؤں میں جامعۃ الصالحات کے نام سے معروف ومشہور ہے، اب تو لڑکیوں کے لیے بڑے بڑے عربی مدارس ادھراُدھر اوربھی کئی ایک ہوگئے اور ہوتے جارہے ہیں لیکن صوری اورمعنوی دونوں اعتبار سے جامعۃ الصالحات کو یک گونہ شرف فضیلت و تقدم حاصل ہے اوریہ سب کچھ نتیجہ ہے مولانا مرحوم کے اخلاص ومحبت،محنت ومشقت اور ذوق تعمیر وحسن انتظام کا۔
راقم نے کئی مرتبہ جامعہ کی دورۂ حدیث کی طالبات کاامتحان لیاہے اور ہرمرتبہ طالبات کے صحیح اور برمحل جوابات سے دل نے مسرت محسوس کی ہے۔ ان کو جامعۃ الصالحات سے عشق تھا، شب وروز اس کے کاموں میں مصروف رہتے تھے۔ اﷲ تعالیٰ ان کوعلماوصلحا کا مقام جلیل عطافرمائے اورجامعۃ الصالحات کوکسی قسم کے شر اورضررسے محفوظ رکھے۔ [جولائی۱۹۸۴ء]

 

Polemic Views about the Source of Qur’ān in Medieval Christian Writings with a Reflection upon Contemporary Orientalists: A Critical Review

Before the advent of Islam, there was a strong tradition of polemic writings both among the Jews and the Christians to prove the errors of adversary. But, after the advent of Islam in general, and the conquering of Roman / Byzantine empire by the Muslims in the era of Righteous Califate in specific, due to embracing Islam by a large number of local populace, the flux of Christian polemic writing was directed towards Islam. A number of polemic writings surfaced as a resort to keep their religion alive. These writings tried to belittle all basic concepts, beliefs, and creeds of Islam, and even the personality of the Prophet Muhammad (PBUH), and the Holy Qur’ān. The contemporary orientalist polemic writers have claimed that there are several accounts originating from Jewish and Christian sources which tried to allegedly prove that the Holy Prophet Muhammad (PBUH) was instructed by Jewish or Christian scholars in the composition of Holy Qur’ān, and to support this claim, they not only point out to certain Jewish or Christian sources, but have parroted their arguments as well, with the similar motives. With this, they have attempted to discredit Islam by raising doubts about the origin of Qur’ān. But despite of their efforts the fact remains firm that the Qur’ān has a Divine origin and was revealed by Allah Ta‘ālā unto Prophet Muhammad (PBUH). In this regard, the purpose of this article is to analyze the medieval polemic writings, their motives, and their rumination by the orientalists of the contemporary age. A critical approach is adopted in this analytical, and historical study, using published authentic data and literature including academic books, research papers, periodicals, dictionaries and reliable web sites also.

Exploring the Co-Relation of a Headteachers Decision Making and Her Prior Professional Experiences in Dealing With Challenges in Intial Years of Headship

This study explores the co-relationship of decision-making and prior professional experiences of a headteacher in dealing with challenges in the initial years of headship in a Government Secondary School in Karachi. The purpose of the study was to get insights into the process of how headteacher's decisions were informed by her prior professional experiences in coping with the challenges in the initial years of headship; and to understand the factors, other than professional experiences which tend to influence a headteacher's decisions in dealing with challenges. It is a case-study of a newly promoted female headteacher with twenty-nine years of teaching experience, posted in the same Government Secondary School. Multiple repertoire of data collection tools (interviews, observations, document analysis and reflective memos) were being used to collect informations about the inquiry. The findings of the study explored the assumption of the study that headteacher's prior professional experiences do influence her decision-making in dealing with the challenges in the initial years of headship. Moreover, it was found that decision-making processes and her prior professional experiences were co-related in teaching and learning, leading and managing the staff, efficient and effective deployment of the staff, strategic direction and development of the school, and resources and accountability. The intensity of the influence, however, varied was found very intense in three major areas of school leadership and management; in teaching and learning, leading and managing the staff and, in effective and efficient deployment of the staff. The findings depicted that a headteacher's decisions were highly dependent on her prior experiences. In the rest of the two areas, strategic direction and development of the school and, resources and accountability data showed that headteacher was dependent on others' professional expertise also. Other factors, which tend to influence the headteacher's decisions in dealing with challenges in the initial years, were found to be: pressure from high management and community, accountability, personal values and priorities, and relationships with stakeholders. The study has proposed recommendations and questions for further exploration.