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Home > Investigation of Spectral Expansions Connected With Schrodingers Operator

Investigation of Spectral Expansions Connected With Schrodingers Operator

Thesis Info

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Author

Zakaria, Kamran

Program

PhD

Institute

Federal Urdu University of Arts, Science and Technology

City

Islamabad

Province

Islamabad

Country

Pakistan

Thesis Completing Year

2015

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/13517/1/Kamran_Zakaria_Maths_2015_FUUAST_Karachi_22.06.2016.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726509250

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The results (1)-(2) are proved in the present research work. (1) The estimation for the solution of the Schrödinger operator in a closed domain. (2) Problem of uniform convergence of spectral expansions connected with Schrödinger operator of continuous functions in a closed domain. The summability and spectral expansions of the first boundary value problem for Schrödinger operator in closed domain are discussed. The representations used in this research work allow the methods given in [20] and [28] for the spectral expansion and summability of Schrödinger operator in a closed domain. The new estimation and summability connected with one of the Schrödinger operator in a closed domain are derived.
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ڈاکٹر ظفرالہدیٰ

آہ! ڈاکٹر ظفر الہُدیٰ
ڈاکٹر ظفر الہدیٰ ایم، اے۔ پی، ایچ۔ ڈی، علامہ شبلیؒ کی بڑی پوتی کے شوہر تھے، ان کا آبائی وطن تو اعظم گڑھ ضلع ہی میں تھا، مگر ان کے گھر کے لوگ دربھنگہ (بہار) منتقل ہوگئے تھے، اس لئے پٹنہ یونیورسٹی میں اپنی انگریزی تعلیم کی تکمیل کی، وہاں سے فارسی اور اردو میں ایم، اے کرنے کے بعد ڈھاکہ یونیورسٹی میں لکچرار ہوگئے، وہیں سے پنشن پاکر ڈھاکہ میں مقیم تھے کہ ۷؍ مارچ ۱۹۷۸؁ء کو اﷲ کو پیارے ہوئے، ان کی وفات علامہ شبلیؒ کے خاندان کا ایک المناک سانحہ ہے، وہ اپنے شاگردوں اور یونیورسٹی کے رفقائے کار میں اپنے اخلاق، اخلاص، محبت اور میٹھی زبان کی وجہ سے بہت مقبول تھے، اسی لئے جب بنگلہ دیش میں خونیں انقلاب آیا تو وہاں کی سفاکانہ اور بیرحمانہ خونریزی میں ہر طرح محفوظ رہے، ان کے اور رفقائے کار تو کراچی منتقل ہوگئے، لیکن انھوں نے ڈھاکہ ہی میں رہنا پسند کیا، بنگالیوں نے غیر بنگالیوں کے ساتھ جو بے رحمانہ سلوک کیا تھا، اس کی وجہ سے بنگلہ دیش کے لوگوں کے خلاف ہندوستان کے مسلمانوں میں بڑا سخت ردعمل تھا، اس کو ڈاکٹر ظفر الہدیٰ اپنے خطوط میں یہ لکھ کر دور کرنے کی کوشش کرتے رہے کہ آخیر یہ لوگ بھی مسلمان ہیں، وہاں کے مسلمانوں کے لئے اسلام کا صالح لٹریچر پیش کرنے کی خاطر شبلی اکیڈمی بھی قائم کی، دارالمصنفین کی مطبوعات کو بنگلہ زبان میں ترجمہ کرانے کی علمی مہم شروع کی، ان کا کام کچھ چل نکلا تھا کہ وہ وہاں پہنچ گئے، جہاں ایک روز سب کو جانا ہے۔
ان کو دارالمصنفین اور اس کے وسیلہ سے میری حقیر ذات سے بڑی محبت رہی ہے، جب کبھی ڈھاکہ گیا تو وہ مجھ سے سگے بھائی کی طرح ملے اور خاطر تواضع میں...

الوحدة النصية و علاقتها بالنمط اللغوي الكبير: دراسة نظرية في ضوء اللسانيات النصية

The text is larger pattern of any language; it consists smaller units of a language. The Aristotle was among the earliest who discussed the unity of text, then letter on Arabian scholars especially philosophers started to relate it with Quranic text and analysis it according this concept; even some of them claim that the unity of Quranic text is the only reason in its “Ijaaz”. And more then that there were some others who discussed its different aspects in Arabic poetry and prose in the same time. On the other hand the western scholar studied the same “textual unity” in different ways in modern textual linguistics by defining the “unity” as a larger pattern of text. This work will discuss the all aspects of textual unit and its relations with larger pattern in the light of textual linguistics.

Prevalence of Vitamin D Deficiency in an Ethnic African Urban Obstetric Population in an Equatorial City Hospital

Introduction: Low Vitamin D levels in pregnancy have been associated with an increased risk of gestational diabetes, pre-eclampsia, bacterial vaginosis, preterm birth, disordered fetal skeletal homeostasis, and impaired offspring lung health. As a result, current public health advice in several countries is to universally supplement Vitamin D during pregnancy. Data regarding the Vitamin D levels of the ethnic obstetric population in Kenya are not available; hence recommendations for Vitamin D supplementation have no scientific basis. Objectives of study: This research was carried out to determine the prevalence of deficiency of Vitamin D in ethnic African obstetric patients who had delivered at the Aga Khan University Hospital, Nairobi, Kenya, by measuring levels of twenty-five hydroxyVitamin D (25OHD - the status marker of Vitamin D), noting that levels in pregnancy worldwide vary from 18 - 84%, and local levels are unknown. Methods: This was a cross-sectional prevalence study carried out at the Aga Khan University Hospital in Nairobi, Kenya, with the study population being ethnic African females who had delivered at the facility. This site was chosen as it is an urban hospital found within the equatorial belt, where majority of the population is dark-skinned (higher melanin levels hamper Vitamin D production via skin, the primary method through which Vitamin D is produced in the body). Blood samples were collected from 97 sequentially sampled newly delivered women after informed consent, and the levels of 25OHD were measured. The prevalence of deficiency was thereafter calculated. The Vitamin D levels were also related to the patient’s Body Mass Index, and their categorical amount of sun exposure. Results: The prevalence of Vitamin D deficiency in the stated population was calculated to be 79.4% (95% CI 70.5 - 86.6%). No association was noted between the Vitamin D levels, and the Body Mass Indices of the participants, and neither was an associated relationship inferred between Vitamin D levels and the sun exposure times. Conclusion: It is concluded that the prevalence of Vitamin D deficiency is present in a significant proportion of the ethnic African urban mothers that choose to deliver at the Aga Khan University Hospital in Nairobi, Kenya. This figure is noted to be amongst the upper part of the range signifying Vitamin D deficiency in pregnancy the world over (18-84%). Recommendations: Further to this finding, the study provides evidence for routine Vitamin D supplementation during pregnancy in ethnic African females living in equatorial cities. Larger