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On the Algebra of Newton Interpolating Series and its Applications

Thesis Info

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Author

Haider, Azeem

Supervisor

Ghiocel Groza

Program

PhD

Institute

Government College University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2004

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/handle/123456789/427

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726805127

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By means of a sequence S of elements of a field K, we defined a K-algebra KS [[X]] of formal series called Newton interpolating series which generalized the formal power series. We study algebraic properties of this algebra and in the case when S has a finite number of distinct elements we prove that it is isomorphic to a direct sum of a finite number of known algebras. A representation of strictly convergent power series as convergent Newton interpolating series is given. Then this representation is used to study problems of the zeros of strictly convergent power series and to solve an interpolation problem. We also study the problem of the zeros of bounded Newton interpolating series. A method for p-adic analytic continuation by means Newton interpolating series is presented.
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حد کا اصطلاحی مفہوم

حد کا اصطلاحی مفہوم
ایسے امور جن کی حلت و حرمت اللہ تعالیٰ نے خود بیان فرما ئی ہے اور ان سے تجاوز کرنے سے منع فرما دیا ہے۔ یہ امور " حدود اللہ " کہلاتے ہیں۔ اہل علم نے حد کی اصطلاحی تعریفیں مندرجہ ذیل بیان کی ہیں:
ایسے کام جن کی حلت و حرمت اللہ تعالیٰ نے بیان فرمائی ہو، جیسا کہ ابن منظور افریقی تحریر کرتے ہیں
"وحُدُود الله تعالى الأَشياء التي بيَّن تحريمها وتحليلها وأَمر أَن لا يُتعدى شيء منها۔"8
"حدود اللہ سے مراد ایسی اشیاء ہیں کہ جن کی حلت و حرمت اللہ تعالیٰ نے بیان فرما دی ہے اور یہ حکم دیا ہے کہ ان سے آگے نہ بڑھا جائے۔ "
علامہ زبیدی ؒ (م:1205ھ)حدود کی اقسام اور اس کا مفہوم بیان کرتے ہیں
"فَحُدُودُ ا الله عزّ وجلّ ضَرْبَانِ : ضَرْبٌ منها حُدودٌ!حدَّها للنّاسِ في مَطَاعِمِهم ومَشارِبِهم ومَنَاكِحِهِم وغيرها ممّا أَحَلّ وحَرَّم،وأَمَرَ بالانتِهَاءِ عمّا نَهَى عنه منها ونَهَى عن تَعَدِّيهَا ، والضَّرْب الثانِي عُقوباتٌ جُعِلَتْ لمنْ ركِبَ ما نَهَى عنْه ، كحَدّ السّارِق۔"9
"حدود اللہ کی دو اقسام ہیں: ایک تو ایسی حدود جو لوگوں کے لیے ان کے ماکولات ، مشروبات اور مناکحات وغیرہ میں بسبب حلال اور حرام متعین کی گئی ہیں یہ ان اشیاء سے رکنے کا سبب ہیں جن سے تجاوز کرنے سے روکا گیا ہےاور دوسری قسم وہ سزائیں ہیں جو ممنوع کام کرنے والوں کو دی جاتی ہیں جیسا کہ چور کی حد ۔ "
امام سرخسی ؒ (م:483ھ)حد کی اصطلاحی تعریف کرتے ہوئے لکھتے ہیں
"في الشرع الحد اسم لعقوبة مقدرة تجب حقا لله تعالى ولهذا لا يسمى به التعزير لأنه غير مقدر ولا يسمي به القصاص لأنه حق العباد وهذا لأن وجوب حق العباد۔"10
"شریعت میں حد اس مقررہ سزا کا نام ہے جو بطور حق اللہ...

میاں بیوی میں سے ایک کا قبول اسلام

In the early days of Islam, either of the married couple entering into Islam does not affect their marriage contract and it continued to be valid even after the Emigration of the Holy Prophet (peace and mercy be upon him) to Madina. In fact, it has been observed until Hudhabiya Truce was signed. It was also included the terms if someone from Quraish without accompanying the wali approaches the Holy Prophet (Peaceand Mercy be upon him), he/she will be returned to Makkah. After this agreement, many women came to Madina and embraced Islam. But their spouses and relatives followed them their way to Madina. They claimed them back to Makkah. In this respect, Allah almighty revealed a verse of Surah Mumtahina, which is an express evidence that such believing women must not be returned to their former infidel husbands. They were commanded so to marry believing husbands after their separation from their disbelieving spouses. Through this verse, Muslim husbands were forbidden to stay and have conjugal relations with their nonbelieving wives, too. Similarly,  every believing wife was forbidden to reside with her disbelieving husband. The companions abided by the ruling in its entirety and separated from their non- believing spouses. Jurists have derived many instructions from the verse 10 of Surah Mumtahina, which are discussed in this paper.

Analysis of Cytochrome Oxidase-I Gene in Tilapia Fish Species Tilapia Zillii

DNA barcoding is a technique concerned with the classification of things that is done with the help of small gene or DNA sequence with a known location on a chromosome that can be used to identify individuals or species in organisms mitochondrial DNA (mt DNA) which is helpful in recognition of meticulous species. It uses sequence variety in a 658-base pair fragment near the 5? end of the mitochondrial cytochrome c oxidase subunit I (COI) gene as a means for species recognition. DNA barcoding is a more defined and steady method in comparison with the classification based on the form and structure of the organism. It is equally useful in any stage of life cycle of fishes. The present research project was designed to recognize tilapia fish species (Tilapia zillii) of Pakistan genetically. Blood samples (n=30) were collected from Tilapia fish and genomic DNA was extracted and confirmed by 1% agarose gel. A short segment of COI gene (680bp) was amplified. PCR products were sequenced and analyzed by bioinformatics tools. Number of the haplotypes was 7. The haplotype diversity was Hd: 0.584 while nucleotide diversity was Pi: 0.00244. The mean intraspecific K2P genetic distance was 0.019. The estimated transition/transversion bias R was 1.40 that showed that this species possess very low genetic diversity. COI may supply a landmark for the classification of associated species at molecular level. As tilapia is extensively used for food and many other purposes so use of DNA barcoding technique is very helpful in discriminating it from other correlated fish species. It will also reduce the chance of mislabeling of tilapia fish species during its trading internationally as well as the species assessment at national level.