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Stanley Depth and Sequentially Cohen- Macaulay Lexsegment Ideals

Thesis Info

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Author

Ishaq, Muhammad

Program

PhD

Institute

Government College University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2007

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/handle/123456789/1637

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727233981

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In the first chapter we give some basic definitions from commutative algebra. We give some results obtained in recent years for the Stanley depth of multigraded S-modules, where S = K[x1 , . . . , xn ] is a polynomial ring in n indeteminantes with coefficients in a field K. We also give some results regarding the progress towards the Stanley’s conjecture. In the second chapter, we show that if I ⊂ J be monomial ideals of a polynomial algebra S over a field. Then the Stanley depth of J/I is smaller or equal to the √ √ Stanley depth of J/ I. We give also an upper bound for the Stanley depth of the intersection of two primary monomial ideals Q, Q , which is reached if Q, Q √ √ are irreducible, ht(Q + Q ) is odd and Q, Q have no common variables. These results are proved in my paper [23]. In the third chapter, we give different bounds for the Stanley depth of a monomial ideal I of a polynomial algebra S over a field K. For example we show that the Stanley depth of I is less than or equal to the Stanley depth of any prime ideal associated to S/I. Also we show that the Stanley’s conjecture holds for I and S/I when the associated prime ideals of S/I are generated by disjoint sets of variables. These results are proved in my paper [24]. In the forth chapter, we give an upper bound for the Stanley depth of the edge ideal I of a k-partite complete graph and show that Stanley’s conjecture holds for I. Also we give an upper bound for the Stanley depth of the edge ideal of an s-uniform complete bipartite hypergraph. In this chapter we also give an upper bound for the Stanley depth of the edge ideal of a complete k-partite hypergraph and as an application we give an upper bound for the Stanley depth of a monomial ideal in a polynomial ring S. We give a lower and an upper bound for the cyclic module S/I associated to the complete k-partite hypergraph. These results are proved in our papers [26] and [27]. In the fifth chapter, the associated primes of an arbitrary lexsegment ideal I ⊂ S are determined. As application it is shown that S/I is a pretty clean module, therefore, S/I is sequentially Cohen-Macaulay and satisfies the Stanley’s conjecture. These results are proved in my paper [25].
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ا یڈو رڈ ویسٹر مارک

مارک، ایڈورڈ ویسٹر
ایڈورڈ ویسٹر مارک کا انتقال
ایڈورڈ ویسٹر مارک نے جو ایک عرصہ تک لندن یونیورسٹی میں عمرانیات کا پروفیسر تھا، گزشتہ اکتوبر میں انتقال کیا، وہ ایک عمیق النظر فلسفی اور بین الاقوامی شہرت کا مالک تھا، اوائل زندگی میں اس کو خیال پیدا ہوا کہ اخلاقی خیالات اور مذہبی اعتقادات میں گہرا تعلق ہے، چنانچہ اس نقطہ نظر سے اس نے یورپ کے علاوہ دوسری قوموں کے روایات اور اعتقادات کا مطالعہ کرنا چاہا اور اس کے لئے مراکش کو منتخب کیا، اور وہاں جاکر چار سال تک مقیم رہا، یہاں اس نے نہ صرف انسانیاتی معلومات جمع کئے، بلکہ وہاں کے لوگوں طرز زندگی و غور و فکر سے بھی واقفیت حاصل کرکے ان کے رسم و رواج کو آسانی سے سمجھا جو تمدن کے مختلف دور میں پیدا ہوتے رہے، یہاں کی تحقیقات اس نے اپنی مشہور کتاب ’’اخلاقی تخیلات کی ابتدا اور نشوونما‘‘ (The origin and development of moral ideas) میں قلمبند کی جو دو جلدوں میں ۱۹۰۶؁ء اور ۱۹۰۸؁ء میں شائع ہوئی تھی، یہ کتاب بہت اہم اور بلند پایہ سمجھی جاتی ہے اور اخلاقیات پر ایک فلسفیانہ مقالہ یا اخلاقی خیالات کی تاریخ کہی جاسکتی ہے، اس کتاب کی خوبی یہ ہے کہ اس میں تحلیلی اور تاریخی تجزیوں میں امتزاج پیدا کرکے یہ دکھایا گیا ہے کہ اخلاقی پسندیدگی اور ناپسندیدگی کی ابتدا معاشرتی تکدر اور تشکر سے ہوتی ہے، ڈاکٹر ویسٹر مارک کی اور دوسری تصنیفات یہ ہیں، مراکش میں انسانی شادی، رسم اور اعتقاد کی تاریخ، (۲)اخلاقی اضافیت، (۳)عیسائی مذہب اور اخلاق۔
(’’ص ۔ ع‘‘، دسمبر ۱۹۳۹ء)

 

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