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Analog of Groebner bases in a polynomial ring with countably infinite indeterminates

Thesis Info

Author

Jabeen, Nazia

Program

MS

Institute

Institute of Business Administration

Institute Type

Private

City

Karachi

Province

Sindh

Country

Pakistan

Thesis Completing Year

2018

Page

30

Subject

Mathematics

Language

English

Other

CallNo: 512.4

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676720937688

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In this dissertation we have developed the theory of Sagbi bases and Sagbi Groebner bases in a polynomial ring with countably infinite indeterminates over a field and also explored the behavior of Sagbi Groebner bases for ideals in algebras of polynomial ring with finite indeterminate over a field under (de)homogenization technique. We have discussed that the things work in a same manner for Sagbi Groebner bases as in case of Groebner and Sagbi bases
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ام المومنین حضرت عائشہؓ

ام المومنین حضرت عائشہؓ
اعتراض نمبر۱۰۷
حضرت عائشہ ؓ کی عمر کے بارے میں کتب و احادیث میں چند ایسی روایات ہیں جن میں آپ کی عمر بوقت نکاح چھ سال اور رخصتی کے وقت نو سال کا ذکر ہے ۔ جس طرح بخاری شریف جلد نمبر ۳ کتاب النکاح ص ۹۴ پر یہ روایت ہے ’’ حضرت عائشہ ؓ فرماتی ہیں کہ نبی ﷺ نے نکاح کیا ان کے ساتھ جب وہ چھ سال کی تھیں اور داخل کی گئیں آپ ﷺ پر (زفاف کے لیے ) جب وہ نو سال کی تھیں اور نو سال تک آپ کے پاس رہیں ۔ ‘‘ (۲) سر ولیم میور لکھتا ہے کہ ’’آنحضرتﷺ سے شادی کے وقت سیدہ عائشہ ؓ کی عمر دس گیارہ برس سے زیادہ نہ تھی ۔‘‘
جواب: مستشرقین نے دو مو قعوں پر بی بی عائشہ ؓ کی عمر کو متنازعہ بنایا ہے ۔ اول جب آنحضرت ﷺ کی حضرت عائشہ ؓ سے نسبت طے ہوئی ۔ دوم :جب شادی انجام پذیر ہوئی ۔ بات تو ایک ہی ہے اگر نسبت کے وقت عمر کا درست تعین ہو جاتا تو آگے غلطی کا امکان نہ رہتا بات یہیں ختم نہیں ہوتی کیونکہ کئی مسلم مورخین نے بھی ٹھوکر کھائی ہے شائد انہیں سہو ہوا ہے یا بلا تحقیق لکھتے چلے گئے یہاں تک کہ چوٹی کے محدثین بھی اپنی کتب احادیث میں یو نہی نقل کرتے ہیں ۔ مثال کے طور پر بخاری شریف میں پانچ ،مسلم شریف میں چار ،اور ابو دائود میں ایک روایت ہے ۔ان روایات کو قبول کر کے مقدس و محترم ہستیوں کی شان میں گستاخی و بے ادبی کرتے رہے ۔ آئیے دیکھیں کہ حقیقت کیا ہے ؟۔
دلیل اول : امام بخاری کہتے ہیں جب قرآن کریم کی ۵۴ ویں سورہ القمر نازل ہوئی...

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