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Hyperplane Arrangements

Thesis Info

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Author

Nazir, Shaheen

Supervisor

Alexandru Dimca

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/123456789/1951

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726384529

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Let A = {H 1 , . . . , H l } be a hyperplane arrangement in C n and M be the complement of the union of hyperplanes in A, i.e., M = C n \ ∪ li=1 H i . The cohomology algebra H ∗ (M, C) has a complete combinatorial description. Let L be a local system on M and H ∗ (M, L) be the cohomology algebra with local coefficients. For [ω] ∈ H 1 (M, C), there is a chain complex: μ ω μ ω μ ω 0 → H 0 (M, C) → H 1 (M, C) → · · · → H n (M, C) → 0. The characteristic varieties of M are the jumping loci of the cohomology groups H ∗ (M, L). The resonance varieties of M are the jumping loci of the cohomology groups of the above complex. The aim of this thesis is to study some properties of these varieties.
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غارِ حرا

غار حرا
مکہ معظمہ سے تین میل کے فاصلہ پر ایک غار ہے جس کو حرا کہتے ہیں ۔ اب اس کو جبل النور کہتے ہیں ۔ اس غار کا طول چار گز اور عرض پونے دو گز ہے اونچائی اتنی ہے کہ ایک دراز قد آدمی کھڑے ہو کر اس میں نماز پڑھ سکتا ہے دھوپ اور بارش سے بھی بہت حدتک محفوظ ہے۔ آپ ﷺْ کھانے پینے کا سامان لے کر وہاں چلے جاتے اور جب تک کھانے پینے کی اشیاء ختم نہ ہوتیں ، آپ ﷺ واپس تشریف نہ لاتے ۔ آپ ﷺ وہاں عبادت میں مصروف رہتے ۔ یہاں سوال یہ پیدا ہوتا ہے کہ آپ کس قسم کی عبادت کیا کرتے تھے ۔ شرح بخاری عینی میں ہے ۔ ترجمہ ’’ یہ سوال کیا گیا کہ آپ کی عبادت کیا تھی ؟ تو اس کا جواب یہ ہے کہ غور و فکر اور عبرت پذیری ‘‘۔ مولانا شبلی لکھتے ہیں ’’ یہ وہی عبادت تھی جو آپ ﷺ کے دادا ابراہیم ؑ نے نبوت سے پہلے کی تھی۔ ستاروں کو دیکھا تو چونکہ تجلی کی جھلک تھی ، دھوکہ ہوا ، چاند نکلا تو اور بھی شبہ ہوا ، آفتاب پر اس سے بھی زیادہ شبہ ہوا لیکن جب سب نظروں سے غائب ہو گئے تو بے ساختہ پکار اٹھے ’’ اِنّیِ لاَ اُحِبُّ الاٰفِلِینo
انی وجھت ۔۔۔۔ والارض ( الانعام ۔۷۹) ترجمہ( میں فانی چیزوں سے محبت نہیں کرتا میں اپنا منہ اس کی طرف کرتا ہوں جس نے زمین و آسمان پیدا کیا ‘‘۔ ایک مغربی مورخ کار لائل نے آنحضرت ﷺ کی عبادت کی کیفیت اس طرح بیان کی ہے ’’ سفرو حضر میں ہر جگہ محمدﷺ کے دل میں ہزاروں سوال پیدا ہوتے تھے، میں کیا ہوں ؟ یہ غیر متناہی عالم کیا ہے ؟ نبوت کیا...

قرآن كا سات حروف پر نازل ہونا

According to Hadith literature, the Quran is revealed in seven Ahruf, the plural of harf. Ahruf are distinct from Qira'at. This is a very momentous and lengthy topic, indeed, one of the most complicated discussions on the sciences of the Qur’an. It is very difficult to discuss it in full details in this work but the important things about it are being presented in this article. The first problem we face with this Hadith is what is meant by the Revelation of Qur’an on Seven “Ahruf”? We find a great deal of difference of opinion on this subject. Up to thirty five different views have been quoted by Ibn al-‘Arabi and others. Some of the popular views are quoted in this article. The context of these narrations indicates clearly that the word ‘seven’ does not denote an unspecified large number but it denotes the specific numerical value ‘seven’. Hence, in the light of these narrations this view (that seven means more than that) does not hold good and the majority of scholars reject it. In the vast collections of Hadiths, we do not find any mention of difference in the Qur’an other than that accounted for in “ahruf”. How then may we explain differences in reading and “ahruf”? I have not been able to find a satisfactory answer to this confusion with the advocates of this theory.

Analysis of Unsteady Natural Convective Radiating Gas Flow

Due to diversity in nature, fluids are present everywhere in universe. We go through air, a form of fluid. We use water in our everyday life, which is much important fluid for the perpetuity of living organism. We use fluid in the form of gases for breathing and for different activities of everyday life. The whole universe is covered with the invisible layers of gas. A gas flows through conduction, convections and radiations. The flow of such gaseous materials attracted different researchers of mathematical society. The main theme of this Thesis is the investigation of the behavior of thermal diffusion in the radiating flow of gases. Flow is studied in an open ended channel, which is stationary and having uniform temperature. Fluid is gradually moving under the effect of temperature. We used Laplace transform for the solutions of non dimensional fractional governing equations of radiating flow. Moreover Caputo time fractional derivative have been used for the dealing of temporal derivative. Closed form analytical solutions for Velocity field and thermal expansions are expressed as Robotnov function, Wright and Hartley function. The effects of factional order parameter α, Prandtl number Pr, Grashof number Gr, Radiative parameter R are examined by the graphical interpretations. It is observed that small value of time t has a role activator at the fluid velocity for increasing values of factional order parameter α. Where, large value of time ’t’ inhibits the flow of fluid for increasing values of factional order parameter α. Large values decreases velocity but the effect of large value of time with increasing value of Prandtal is not significant. Increased in value of radiating parameter R decrease the speed of gas and finally Grashof number Gr has direct relation with velocity. The fractional differential equation for temperature distribution operated by Laplace transform. The partial differential equation of velocity is solved numerically with nanoparticle namely copper with the water as based fluid by using Stehfests algorithm. The effects of fractional order derivative and physical parameters are graphically investigated.