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Combinatorial and Arithmetic Study of Binomial Edge Ideal

Thesis Info

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Author

Zafar, Sohail

Program

PhD

Institute

Government College University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2008

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/2049/1/1703S.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676725745418

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Let J G denote the binomial edge ideal of a connected undirected graph G on n vertices. This is the ideal generated by the binomials x i y j −x j y i , 1 ≤ i < j ≤ n, in the polynomial ring S = K[x 1 , . . . , x n , y 1 , . . . , y n ] where {i, j} is an edge of G. Our aim in this thesis is to compute certain algebraic invariants like dimension, depth, system of parameters, regular sequence, Hilbert series and multiplicity of J G of some particular classes of binomial edge ideals of graphs. A large amount of information of an ideal is carried by its minimal free resolution. So we give information on the minimal free resolution on certain binomial edge ideals. We also give a complete description of the structure of the modules of deficiencies of binomial edge ideals of some classes of graphs. A generalization of the concept of a Cohen-Macaulay ring was introduced by S. Goto [7] under the name approximately Cohen-Macaulay. In this thesis we collect a few graphs G such that the associated ring S/J G is approximately Cohen-Macaulay. We also characterize all the trees that are approximately Cohen-Macaulay. As more generalized notion than approximately Cohen-Macaulay we also study se- quentially Cohen-Macaulay property for binomial edge ideals. We give a nice con- struction principle in this topic. ̃ on n vertices has the property that S/J̃ is a Cohen-Macaulay The complete graph G G domain with a 1-linear resolution. As one of the main results we clarify the structure of S/J K m,n , where K m,n denotes the complete bipartite graph.
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قائم دین

                قائم دین (افسانوی مجموعہ)

                قائم دین ناطق کا پہلا افسانوی مجموعہ ہے جسے 2010ء میں آکسفورڈ اور پھر 2018ء میں سانجھ پبلی کیشنز نے چھاپا یہ کتاب 128 صفحات پر مشتمل ہے اور اس کا انتساب افتخار عارف کے نام ہے۔یہاں  ہر ایک افسانے کا الگ الگ جائزہ پیش کرنا مشکل ہے۔

                مصنف مذکور نے مشاہدے کی کیا صورت حال  بیان کی ہے۔اس کا طرز بیان کیسا ہے اس کے مطابق یہ بتانا مقصود ہے کہ جہاں انہیں زبان پر عبور حاصل ہے وہاں ان کا طرز بیان بھی نہایت سادہ اور جلد سمجھ آجانے والا ہے۔قاری کو اپنی گرفت میں رکھتا ہے۔ قائم دین کے تمام افسانوں میں ان کا ایک خاص انداز تحریر ہیکہ قاری ایک ہی نشست میں تمام افسانے پڑھ لینا چاہتا ہے۔ ہر افسانہ اپنے اندر ایک منفرد کہانی رکھتا ہے،قاری افسانے میں پڑھنے والے واقعات کو اپنی زندگی کا حصہ سمجھتا ہے۔ظفر اقبال ناطق کے افسانوں پہ انھیں  داد دیتیہوئے لتین  ہیں کہ:

ہم تو جو ہیں جہاں سے آئے ہیں

آپ لیکن کہاں سے آئے ہیں

 

                ناطق کا کام دادکے قابل ہے جو انھیںدوسرے ادیبوں سے ممتاز کرتا ہے ہر افسانہ اپنے اندر ایک امتیازی وصف رکھتا ہے۔افسانوں میں موجود تمام کہانیاں حقیقت سے اس قدر قریب ہیں کہ قاری کو اپنی زندگی کا حصہ معلوم ہوتی ہیں۔

احکام القرآن میں اسلوب تفسیر اور امام جصاص پر نقد: تجزیاتی مطالعہ

Imam Abu bakar al-jassas is famous jurist of Hanafi school of thought. He wrote a great interpretation of Quran named Ahkam ul Quran. He has written more than fifteen books. In his interpretation of Quran, he doesn’t interpret all Qur’anic verses, as is done by other commentators. He just explains the verses, which have directly a connection with fiq'h ul Ahkam. Al-jassas usually expresses his teachers as al-sarkhasi and other Hanafi's preferred opinion, and always gives systematic reasons for his preference. In the beginning of this article the methodology and characteristics of this interpretation are described in detail. There are many features in Ahkam ul quran that we have expressed i.e. Ahkam ul Quran is based on conventional narration, authentic quotations from the Islamic scholars and lingual and grammatical discussion with a critical explanation of Qur’anic words from lexical to technical and connection on the basis of grammatical and syntax regulations and illustration of differences between synonyms. He had been blamed by some traditionalists that he interprets some verses in wrong perspective and he discussed many irrelevant points in this. This article is an analytical study about these allegations. The result of this study is that kind of attitude towards his book is quite wrong by nature because imam Al-jassas describes his opinion in the light of his school of thought. Although, he got an Aggressive Behavior towards some famous Personalities of Islam, which is Inappropriate. In the end of study, the paper concludes that nobody can deny the importance of his interpretation at all. 

Investigation on the Antioxidant and Antimutagenic Potential of Selected Indigenous Medicinal Plants.

The present study has been designed to investigate the antioxidant and antimutagenic potential of hydrolyzed and non-hydrolyzed extracts of twenty medicinal plants belonging to different plant families like Amaranthaceae, Cucurbitaceae, Labiatae, Fabaceae, Solanaceae, Poaceae, Malvaceae and Zygophyllaceae. Hydrolyzed extraction was carried out using acidified methanol and non-hydrolyzed extracts were obtained with aqueous methanol. Both types of extracts were evaluated for phenolic contents, antioxidant, mutagenic and antimutagenic potential and genoprotective activity. Hydrolyzed extracts of all plants represented higher phenolic contents, antioxidant and antimutagenic potential as compared to non-hydrolyzed extracts. Mentha longifolia (402.8±1.9 mg GAE/g dry extract), Citrullus colocynthus (396.4±1.9 mg GAE/g dry extract) and Mimordica dioca (354.4±1.9 mg GAE/g dry extract) showed highest phenolic contents. Cassia angustifolia (230.4±2.6 mg AAE/g dry extract), Mimordica dioca (132.6±0.8 mg AAE/g dry extract) and Mentha longifolia (131.7±1.5 mg AAE/g dry extract) presented largest reducing power in terms of ascorbic acid equivalents (AAE). Highest free radical scavenging was shown by Citrullus colocynthus (96.21±1.0%; IC50 2.11±0.12 µg/mL), Achyranthus aspera (93.1±0.4%; IC50 6.5±0.10 µg/mL), Mentha longifolia (92.4±1.0%; IC50 14.3±0.1 µg/mL). Citrullus colocynthus (96.21±1.0%; IC50 2.11±0.12 µg/mL), Mimordica dioca (91.91±1.0%; IC50 17.22±0.36 µg/mL) and Mentha longifolia (92.4±1.0%; IC50 14.3±0.1 µg/mL) exhibited highest antioxidant activity in terms of inhibition to lipid peroxidation. Lowest cytotoxicity was observed for Citrullus colocynthus (2.65±0.09%), Achyranthus aspera (2.95±0.14%) and Mentha longifolia (3.95±0.08%) by evaluating haemolytic activity. Highest antimutagenic potential was calculated for hydrolyzed extracts of Citrullus colocynthus (75.79±0.69%), Mimordica dioca (73.02±1.37%) and Achyranthus aspera (72.22±0.69%). Also, Citrullus colocynthus, Mimordica dioca and Achyranthus aspera showed considerable protection against oxidatively damaged DNA plasmid i.e. pBR322. Finally, it has been concluded that Citrullus colocynthus, Mentha longifolia, Mimordica dioca and Achyranthus aspera were the most bioactive plants of all selected medicinal plants as represented by the results of LCMS analysis that exhibited the presence of high concentrations of bioactive compounds in the extracts of these plants.…………………………………