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Soft Rough Fuzzy Graphs With Applications

Thesis Info

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Author

Zafar, Fariha

Program

PhD

Institute

University of the Punjab

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2019

Thesis Completion Status

Completed

Subject

Physical Sciences

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/11167/1/Fariha%20Zafar_Maths_2019_UoP_Punjab_26.03.2019.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676725034659

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Vagueness arises in several complex issues of engineering, science and many other fields. These issues cannot be resolved using crisp mathematical methods. Rough set theory is an approach for dealing with the inaccurate and ambiguous knowledge. Because of the confinement of human’s learning to comprehend the perplexing issues, it is exceptionally hard to apply one kind of uncertainty technique to address such issues. Consequently, it is important to build hybrid models by combining the benefits of numerous other diverse mathematical models dealing with uncertainty. In this dissertation, the concepts and methods of fuzzy sets, rough sets and soft sets are applied to construct rough fuzzy digraphs and soft rough fuzzy digraphs. Several new concepts, concerning these digraphs are introduced. Also various properties of these two models are discussed in detail. Some real world problems are also presented and solved through these graphical models. Furthermore, efficient algorithms are designed to solve these problems and the time complexity of these algorithms are also computed.
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8۔عدالتی فیصلوں میں تاخیر

پاکستان جیسے ملک میں پولیس کی سستی اور عدم تعاون کا مرکز عمومی طور پر حدودمقدمات ہوتے ہیں ۔ پھر ان مقدمات کا دعویٰ بہت تاخیر سے دائر ہوتا ہے ۔ پھر بہت زیادہ تاخیر سے ان مقدمات کے کیس عدالتوں میں لگتے ہیں اور بآلاخر ان مقدمات کے فیصلے بھی بہت سالوں بعد ہوتے ہیں ۔اس کی بنیادی وجہ پولیس کی سستی ، رشوت کی عادات ، وکلاء کا عدم تعاون اور ججز کی تعداد کا کم ہونا ہے۔ پاکستانی معاشرہ اس وقت جرائم کا گڑھ بن چکا ہے ۔ روزاانہ کی بنیاد پر کافی زیادہ جرائم رپورٹ بھی ہوتے ہیں اور ان کی FIR بھی درج ہو تی ہیں ۔پھر یہ مقدمات عدلیہ کے سامنے پیش ہو تے ہیں ۔ اس طرح حدود وقصاص کے مقدمات سالوں تک چلتے ہیں۔ اس کے علاوہ اداروں کے آپس میں مربوط نہ ہونے کی وجہ سے حدود وقصاص کے مقدمات کو غلط طریقے سے نمٹایا جا تا ہے۔ وقت پر انصاف نہ ملنے کی وجہ سے مظلو م اور متضرر اپنے عدالتی نظام سے اور انصاف نہ ملنے پر ریاست سے بھی بد ظن ہو جاتے ہیں ۔

شاہ ولی اللہ کی قرآنی خدمات کا تحقیقی جائزہ

Hazrat Shah Waliullah (R.A), a well-known scholar, was born in Delhion the 21st of February, 1703 A.D. He rendered many religious, political, economic and social services to The Muslims of India. In many of his services, the greatest one is that he translated The Holy Quran into Persian becausemany people were unfamiliar with Arabic. Later his sons, Shah Abdul Qaderand Shah Rafi-ud-Din, translated The Holy Quran into Urdu so more and morepeople could gain Islamic knowledge. He elaborated The Quranic sciences indifferent aspects and the research in your hand is also about his “QuranicServices”. He wrote fifty one books in Persian and Arabic. Amongst the mostfamous were Hujjatulla-hil-Baligha and Izala-tul-khifa. He died on 20th ofAugust, 1762 A.D in Delhi, India.

Understanding How Grade - 6 Government School Mathematics Teachers Can Promote in - Depth Student Learning in Mathematics by Using Experiences from Students Daily Lives

In Pakistan, including developing countries, learning Mathematics in schools is considered to be different from using it in daily life situations primarily because approaches to teaching and learning Mathematics sometimes do not encourage students to see connections between the teaching learning Mathematics and daily life situations. Therefore, students often find it difficult to solve their daily life mathematical problems in spite of studying Mathematics for several years in schools. In order to explore the question “How can the teacher relate the teaching of Mathematics to students’ daily life experiences (DLE)?” I conducted a study embedded in qualitative inductive research paradigm, in one of the secondary schools in the public sector in Karachi, working with two Mathematics teachers and 17 students of Grade-6. The study consisted of three phases (entry negotiation, pre-intervention, and intervention). The main purpose of the study was to find out how concrete experiences of DLE can make mathematical learning less abstract and help students see connections between learning of Mathematics in schools and its use in their daily life situations. The on-going process of data analysis and interpretation was based on the pre-determined major themes related to the research questions, followed by sub themes emerged from the collected data. The study identifies some of the major issues concerning the teaching of Mathematics through students’ DLE. One of the findings suggests that the use of students’ DLE in the teaching of Mathematics as concrete mathematical representation helps them learn Mathematics with understanding. On the basis of the findings, this study puts forward some recommendations in the relation to the teaching of Mathematics. With additional professional support, teachers can design mathematical activities from students’ DLE such as shopping, sports and so on. These activities can be used to relate the teaching of Mathematics to students’ DLE ensuring the relational mathematical understanding.