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Stability Estimates for the Elliptic and Parabolic Obstacle Problems

Thesis Info

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Author

Naveed Ahmad

Program

PhD

Institute

Government College University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2006

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/handle/123456789/1783

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727230873

Similar


Stability of the solution of the multidimensional unilateral obstacle problem with respect to the variations of the coefficients of the corresponding second order partial differential operator is studied in the present thesis. A H ̈lder type stability estimate in the second order Sobolev space of functions is o established for the solution of the elliptic obstacle problem. Rodrigues ([25, 1987]) proved the stability result with respect to bounded external force functions under the nondegeneracy condition. We generalize this result to the case of arbitrary functions belonging to the space Lp (D), p ≥ 2. Further parabolic obstacle problem is studied with different principal parts in the parabolic differential operator and applied to the American put option problem with various local volatility functions. The estimate of the area between the early exercise boundaries through the uniform distance between the local volatility functions is obtained.
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پروفیسر منظور حسین شورؔ

پروفیسر منظور حسین شورؔ 
(ڈاکٹر غلام مصطفےٰ خان)
شور صاحب (اﷲ بخشے) میرے دیرینہ کرم فرما تھے۔ ان کا بچپن کا نام منظور علی تھا جیسا کہ میں نے ان کے مکان پر ایک کتاب میں لکھا ہوا دیکھا تھا۔ بعد میں ان کا نام منظور حسین ہوا۔ دوھیال ایچپور کی تھی اور ننھیال اکولہ کی تھی۔ یہ دونوں شہر برار میں ہیں، وہ امراؤتی (برار) کی شہر پناہ کے ناگپوری دروازے کے قریب ایک آبادی میں جو سادات کی تھی دسمبر ۱۹۱۰؁ء میں پیدا ہوئے۔ والدضامن علی صاحب جو بعد میں کراچی آکر ۱۹۶۸؁ء میں فوت ہوئے، تھانیدار تھے۔ بہت سیدھے سادے تھے، امراؤتی میں بارہا ان سے شرفِ ملاقات حاصل ہوا تھا، شور صاحب کی ابتدائی تعلیم امراؤتی ہی کے محمڈن اسکول میں ہوئی، اس زمانے میں میٹرک کی گیارہویں جماعت ہوا کرتی تھی، یہ اسکول جس کا نام اب تبدیل کردیا گیا ہے مال ٹیکری کے قریب ہے۔ اور اب اس ٹیکری پر شیوا جی کا مجسمہ نصب کردیا گیا ہے۔ شور صاحب نے ۱۹۲۸؁ء میں وہاں سے میٹرک پاس کیا۔ پھر علی گڑھ تشریف لے گئے۔ وہاں میرس ہوسٹل میں ان کا قیام تھا۔ ناگپور کے مونس حسین ان کے خاص دوست تھے، علی گڑھ کے انٹرمیڈیٹ کالج میں اس وقت نویں دسویں گیارہویں اور بارہویں جماعتیں تھیں، میرس ہوسٹل، ارون سرکل یانیوسرکل کے چار ہوسٹلوں میں سے ایک تھا۔ اس کے علاوہ منٹو سرکل میں ان طلبہ کے لیے چار ہوسٹل تھے اور وہاں دو ہوسٹلوں (اے۔بی) میں تعلیم بھی ہوا کرتی تھی۔ ڈے اسکالر اور سیمی بورڈ ان کے علاوہ تھے۔ مولانا احسن مارہروی مرحوم کی وجہ سے طلبہ میں شعر و شاعری کا ذوق زیادہ پیدا ہوگیا تھا۔ وہ طرحی مشاعرے بھی منعقد کراتے تھے اور کل ہند مشاعرے بھی انہی کے دم سے قائم ہوئے تھے۔ شور صاحب کی...

Reading in a report: the French National Gendarmerie, on the reasons for the cargo train coup between the cities of Ain Sefra and Bashar in 1957

It is no coincidence that the researcher in history, especially the history of the Algerian revolution, finds a missing link in the history of the Algerian revolution, which we call the history of what history has neglected. Intentionally, or unintentionally, the archives of Algeria in France, Pax- Provence, or others, or even with some Algerian families, who still hide their archives from researchers? a question that the Algerian researcher will answer to show the truth with documents. Similar to this document, which deals with a topic, in which scientific material was scarce, which is related to the revolution’s strategy to abort and eliminate French projects, the strategy of the National Liberation Army to adapt to the situation consisted in sabotaging French transportation routes, especially railways, in order to destroy France’s economy as well as When she wanted to isolate the people from their revolution, by building barbed wire at the level of the East and West.

Expression of Circadian Clock and Cell Cycle Genes in Chronic Lymphocytic Leukemia

Circadian rhythms are endogenous, self-sustained oscillations of multiple biological processes with approximately 24-hr rhythmicity. Circadian genes and their protein products constitute the molecular components of the circadian oscillator that form positive/negative feedback loops and generate circadian rhythms. The circadian regulation extends beyond clock genes to involve various clock-controlled genes (CCGs) that include various cell cycle genes. Aberrant expression of circadian clock genes could have important consequences on the transactivation of downstream targets that control the cell cycle and on the ability of cells to undergo apoptosis. This may lead to genomic instability and accelerated cellular proliferation potentially promoting carcinogenesis. The current study was carried out to gain further insights into the roles of circadian genes and their downstream targets (cell cycle genes) in chronic lymphocytic leukemia (CLL). We analyzed peripheral blood from 37 CLL patients and equal number of their age- and sex-matched healthy controls for the expression of the four circadian clock and three cell cycle genes. The expression levels of BMAL1, PER1, PER2,MYC, CYCLIN D1 and WEE1 were significantly impaired in CLL cases compared with those in healthy individuals (P < 0.001). BMAL1, PER1, PER2 and WEE1 were found down regulated whereas MYC and CYCLIN D1 were found upregulated. This implies that the deregulated expression of circadian clock genes through their influence on downstream clock-controlled cell cycle genes can play a role in the manifestation of CLL. Moreover, when expression levels of abovementioned genes were compared between shift workers and non-shift workers within the CLL group, the expression levels were more aberrant in shiftworkers compared to non-shift workers. This indicates that circadian dysregulation in terms of shift work may also be a contributing factor in the etiology of CLL. In the current study, serum melatonin levels were also determined in 37 CLL cases and their healthy controls. Serum melatonin levels were found significantly low (P<0.05) in CLL subjects as compared to healthy controls. Furthermore, melatonin levels were found still lower in shift workers as compared to non-shift workers within CLL group. Our results suggest that down regulation of BMAL1, PER1 and PER2 is related to upregulation of Cyclin D1, MYC and down regulation of WEE1 in CLL. Thus, aberrant expression of clock genes can lead to abnormal expression of downstream cell-cycle genes and play a role in the manifestation of CLL. Moreover, low melatonin levels in CLL patients may play a part in xivderegulation of circadian clock gene expression and shiftwork serves as a further contributing factor to an already perturbed circadian clock genes’ expression and low melatonin levels in CLL.