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Parameterization and Graphical Representation of Certain Groups

Thesis Info

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Author

Imran, Tahir

Program

PhD

Institute

Riphah International University

City

Islamabad

Province

Islamabad

Country

Pakistan

Thesis Completing Year

2018

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/10464/1/Tahir%20Imran_Math_2018_Riphah_PRR.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726838921

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The objective of this research study is to investigate certain actions of the group ? de ned by linear-fractional transformations of the form r : z ! z ? 1 z and s : z ! ?1 2(z + 1) , satisfying the relations < r; s : r3 = s4 = 1 >. These actions can be on in niteelds (projective lines over real and imaginary quadraticelds) or onniteelds PL(Fq) for prime q. It has been shown that coset diagram for the actions of ?= G 3;4(2;Z) satisfying the relations < r; s : r3 = s4 = t2 = (rs)2 = (rt)2 = (st)2 = 1 > is connected and transitive on rational projective line. Using this, it is proved that for the group ?, < r; s : r3 = s4 = 1 > are de ning relations. It is also found out that if is any real quadratic irrational number then a unique closed path could be formed by ambiguous numbers in the orbit? . Actions of the group ?on PL(Fq) have been parameterized. It means that for a prime q and 2 Fq, a coset diagram D( ; q) represents each conjugacy class of actions of ?on PL(Fq) where q is a prime number. ln particular, each conjugacy class for actions of in nite triangle groups(3; 4; k) on PL(Fq) is associated with a coset diagram D( ; q).
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پاکستان کی جغرافیائی اہمیت

پاکستان کی جغرافیائی اہمیت
نحمدہ ونصلی علی رسولہ الکریم امّا بعد فاعوذ بااللہ من الشیطن الرجیم
بسم اللہ الرحمن الرحیم
معزز اسا تذہ کرام اور میرے ہم مکتب شاہینو!
آج مجھے جس موضوع پر اظہار خیال کرنا ہے وہ ہے:’’پاکستان کی جغرافیائی اہمیت‘‘
صدرِذی وقار!
پاکستان کا لفظ جہاں فی نفسہ اہمیت کا حامل ہے، وہاں اس کی جغرافیائی اہمیت بھی مسلم ہے۔ اس کے محل وقوع کی اپنی ایک اہمیت ہے، اس کے خدوخال کی اپنی ایک اہمیت ہے، اس کے فضائی علاقے کی اپنی ایک اہمیت ہے، اس کے بری اور بحری علاقے کی اپنی ایک اہمیت ہے۔
جنابِ صدر!
ہمارے ملک پاکستان میں پہاڑی علاقوں کی ایک اہمیت ہے، سیاح خواہ غیر ملکی ہوں یا ملکی پہاڑوں کے دلاویز مناظر ان کے لیے سراسیمگی کے ساتھ ساتھ مسرت و انبساط کا باعث بنتے ہیں، ان کی فلک بوس چوٹیاں آنکھوں کو نور بصارت فراہم کرتی ہیں، ان میں رہائش پذیر جفاکش انسان کاہل اور تساہل پسند انسان کے لیے مہمیز ثابت ہوتے ہیں۔
صدرِ ذی وقار!
دیگر ممالک کے لیے اس کی سرحد آسمان پر دھنک کا منظر پیش کرتی ہے، اس کی وادیوں کو سیراب کرتی ہوئی جوئے نغمہ خواں کا اپنا ایک حسن ہے، اس کی فضائوں میں محو پرواز طائران خوش الحان کی اپنی ایک اہمیت ہے، اس کی آب وہوا کا اعتدال حسن و جمال کا مرقع ہے، اس کی سونا اگلنے والی زمین کی اپنی ایک اہمیت ہے۔
جنابِ صدر!
چین ہو، ایران ہو، افغانستان ہو، یا ہندوستان جملہ ممالک اس کی ہمسائیگی پر خوش ہیں ، چین ، ایران اور افغانستان تو اس کے ساتھ بہتر تعلقات پر نازاں ہیں اور انڈیا اس اسلامی خطے سے محفوظ و مامون ہے۔ کیونکہ اس کے باسی اسلام کی دولت سے مالا مال ہیں جو ایک سلامتی...

عصرحاضر کے تقاضوں کے تناظر میں جامعات دینیہ کا قضیہ: عملی تجاویز

The Muslim religious seminaries (Jameaat-i-diniyya/dini madaris) have become a theme of the global academic agenda, particularly in the wake of the rise of political Islam and the Afghan resistance against Soviet invasion. The theme continuously looms large and has attracted reputable scholars to address the issue in a critical manner. In the present article the author describes various aspects of madrassa education and suggests a number of workable solutions including a new curriculum under the auspices of the Madrassa Education Board and the Higher Education Commission (HEC) in consultation with the traditional madaris and university scholars.

Single Nucleotide Polymorphism-Based Association Studies of Bladder Cancer Patients

There are a growing number of studies conducted in different parts of the world to understand the genetic etiology of urinary bladder cancer (UBC), which is a life- threatening disorder. Therefore to find the susceptible genetic loci we conducted a case-control genetic association study on Pakistani urothelial carcinoma patients (N = 200) and healthy controls (N = 200). For this purpose, four types of sequence variations were studied viz. VNTR polymorphism of eNOS, Alu repeat variation of ACE gene, null polymorphisms of GSTT and GSTM genes and selected common variants of GSTP1, MTHFR, PSCA, TNFα, p21, TP53, CYP1B1, XPD, XRCC1, CAV1, PON1, IGFBP3, VEGFA, LEP, LEPR, PPARγ genes as well as intergenic 8q24 region. In addition to an overall risk assessment, these polymorphisms were also analyzed with respect to the smoking status as well as with respect to tumor grade and stage. Haplotype-based association analysis of variants residing in linkage disequilibrium were also carried out and a gene-gene interaction was studied with reference to combined genotype analysis of functionally related genes. The risk variants of GSTM, LEPR, ACE, PSCA and 8q24.21 locus (rs9642880 and rs6983267) were found to be associated with significantly higher risk while IGFBP3 variant and haplotypes of CAV1 and MTHFR were found to be associated with reduced risk of UBC in the overall comparison of cases and controls. In the gene-smoking interaction CYP1B1, p21 (Ser allele), ACE and rs9642880 conferred a high UBC risk in smokers while LEPR and PSCA variants were found to be associated with increased risk of bladder oncogenesis in non-smokers only. In addition, p21 (Arg allele) was found to be associated with reduced UBC susceptibility in smokers while IGFBP3 and CAV1 haplotypes protected against urothelial carcinoma of the bladder in non-smokers only. GSTM0 and the risk allele of rs6983267 did not show a gene-smoking interaction because of their significant risk contribution in both smoker and non-smoker groups. With reference to tumor grade and stage, a trend of similar genetic etiologies was observed in low grade and non-invasive tumor, while the high grade and invasive tumor types were also found to have common genetic etiologies which were different from the former group. GSTM0, LEPR and rs9642880 were found to be associated with enhanced risk of low grade as well as non-invasive bladder carcinoma. GSTT0, CAV1, PSCA and PPARγ were found to predispose individuals to an elevated risk of ixhigh grade and invasive tumor. ACE and rs6983267 were non-specifically associated with both low and high grades as well as with non-invasive and invasive tumors. IGFBP3 SNP protected against low and high grade as well as against non-invasive stage. The haplotypes of MTHFR were found to confer a high risk of non-invasive tumor while providing protection against MIBC. In brief, the present study revealed the association of some of the genetic variants to the overall disease susceptibility in addition to some gene-smoking and gene-gene interactions.