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Limiting Reiteration for Real Interpolation and Optimal Sobolev Embeddings

Thesis Info

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Author

Irshaad Ahmed

Supervisor

Georgi Eremiev Karadzhov

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/123456789/2037

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726583618

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Firstly, sharp reiteration theorems for the K−interpolation method in limiting cases are proved using two-sided estimates of the K−functional. As an application, sharp mapping properties of the Riesz potential are derived in a limiting case. Secondly, we prove optimal embeddings of the homogeneous Sobolev spaces built-up over function spaces in R n with K−monotone and rearrangement invariant norm into another rearrangement invariant function spaces. The investigation is based on pointwise and integral estimates of the rearrangement or the oscillation of the rearrangement of f in terms of the rearrangement of the derivatives of f .
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نوکر جدوں تیک رہیا ہاں

نوکر جدوں تیک رہیا ہاں
تیرے ہو نزدیک رہیا ہاں
عمر نہیں اینویں ضائع کیتی
تیری وچ اڈیک رہیا ہاں
کرچی کرچی ہوئی روح نوں
درداں نال دھریک رہیا ہاں
اوسے در دا خادم ہاں میں
اوتھوں منگ دا بھیک رہیا ہاں
جس رستے تے مجنوں ٹریا
اوہو لبھدا، لیک رہیا ہاں
غیراں نوں میں دُکھ دِتے نیں
تیرے نال تے ٹھیک رہیا ہاں

الكهنة والمبشرون بقدوم الرسول الكريم محمد (ص) قبل البعثة

الملخص: بعث الله سبحانه وتعالى سيدنا محمداً (ص) رحمة للناس؛ وليرشدهم إلى طريق الهدى، وإلى تصحيح أمور حياتهم، فهو (ص) خير من أُرسلَ لإتمام هذه الرسالة، ونشر دين الإسلام إلى البشرية؛ إذ وكان الإنسان الوحيد في التاريخ الذي نجح نجاحاً مطلقاً على كلا المستويين الديني والدنيوي، فهو من دعا إلى الاسلام ونشره كواحد من أعظم الديانات، وبعد (14) قرناً من وفاته، فأثره مازال باقياً متجدداً لما له الأثر الأعظم في التاريخ، والعالم. وهدفت الدراسة لاظهار مدى معرفة التنبؤات التي جاءت من خلال الكهنة والمبشرين والذي عرفت من خلال الكتب السماوية التي سبقت مجيء الحبيب المصطفى (ص) واستخدم الباحث المنهج الموضوعي في تسلسل الاحداث التاريخية، وقد توصلت الدراسة الى مجموعة من النتائج اهمها: اكدت الدراسة ان تنبئ العديد من الكهنة والعرافين بقدوم محمد (ص) ولكن كان العالم بانتظاره على مدى طويل فنلاحظ ان الكهنة في العصر الجاهلي قد تحدثوا عن نبي يأتي في المستقبل الا انهم كانت رؤيتهم الى النبيالاكرم القادم رؤي واضحة الى اولى الالباب وهذا مااصبح جليأ في ولادة نبي محمد (ص) والاحداث التي رافقته الى بعثتة فكانت مصداقا لاقوالهم انذاك. الكلمات المفتاحية: الكهنة، البشارة، الرسول، محمد (ص)، البيعثة

Association Mapping of Important Traits in Brassica Napus L.

Brassica napus is an important cultivated oilseed crop famous for good quality oil (Canola oil) yet its production is insufficient to meet the domestic demands. Consequently huge foreign exchange is spent on the import of edible oil to meet domestic requirements. One of the problems due to which reduction in yield occurs, is pod shattering at maturity and during harvesting. Enhanced inherent shatter resistance is the best solution for this problem. The phenotypic variation of such complex trait can be studied through association mapping. The present study was based on (i) Diversity analysis of a large population set of Brassica napus, (ii) Delineation of marker-trait associations by using simple sequence repeats (SSRs) markers, (iii) Genome wide association studies for pod shatter using single nucleotide polymorphisms (SNPs) and (iv) Functional profiling of a candidate gene for pod shattering. Four experiments were conducted to achieve these objectives. In the first experiment, germplasm comprising 328 indigenous and exotic lines along with a high yielding check variety (Faisal canola) was evaluated for two consecutive seasons at two locations in Pakistan following augmented layout design in order to determine the magnitude of variation for various quantitative traits. The studied germplasm significantly differed for all the traits, hence useful variability exist which can be helpful in breeding new improved cultivars. Parameters like; number of days taken until flower initiation (DFI), no. of days until 50 % flowering (DF50 %), no. of days until flower completion (DFC), no. of days until 50 % maturity (DM50 %) and no. of days until 100 % maturity (DM100 %) were the major factors that contributed towards diversity of the germplasm hence these traits may be utilized in selection of diverse lines. The genotypes; BN48 303, BN-331, BN-574, BN-304, BN-309 BN-320, BN-306 and BN-310, averaged less than 5% pod shatter in both years. The second experiment was about estimation of population structure and linkage disequilibrium (LD) to determine marker-trait associations in Brassica napus. A collection of 75 randomly selected most diverse lines, based on the results of previous experiment, was genotyped using 55 polymorphic SSRs covering whole genomes. The genotype data were used to assess the population structure and LD through software STRUCTURE v 2.3.4, after removal of minor alleles with P > 0.01, marker trait associations were determined through TASSEL v 4.0. Minimum 1 and maximum 4 SSRs were found to be highly associated with each trait studied. Important marker-trait associations found were days to 100 percent maturity with “Ra2A11”, pod shatter percentage with “sNRA85”, plant height with “sR94102” seed yield per plant with “PBCESSRJU15” and thousand seed weight with “sNRB35”. These highly associated markers after careful validation may effectively be utilised for marker assisted breeding of Brassica napus. The third experiment was carried out in crop genetics department at John Innes Centre (JIC), Norwich, UK during 6 month research placement through IRSIP, HEC Pakistan. It involved genome wide association studies of Brassica napus for pod shatter using SNPs. A collection of 85 diverse Brassica napus genotypes maintained at JIC was used in this experiment. The genomic data were already available (Harper et al., 2012). After removal of minor alleles, 62980 SNPs were used for marker-trait association. Phenotyping of the germplasm was done through random impact test (RIT). Prior to shaking the pod samples were calibrated to 50 % relative humidity at 23°C in order to eliminate the effect of differential 49 pods drying. After each interval of shaking for 8 seconds, broken and intact pods were counted enabling me to draw a decay curve thus half lives were calculated though software R. The genomic data and half life data for both years were then analysed through TASSEL v 4.0 and the results were plotted on a Manhattan plot. Out of 62,980 SNPs, a total of 154 were found to be highly associated with half lives of pods with P < 0.01. These 154 SNPs reside in 110 gene loci in a reference genome of Brassica napus (Harpar et al., 2012) rendering them as possible candidate genes for further studies. The fourth experiment was about profiling the functioning of PIF4-a candidate gene. The reason for considering it a candidate gene for pod shatter was previous study by de Lucas et al.(2008) in which they concluded that interaction between PIF4 transcription factor and DELLA proteins coordinate light and gibberellins responses and the GA/DELLA pathway is involved in controlling seed dispersal (Feng et al., 2008). Hence a number of tests were performed that include RIT of Arabidopsis pif4 mutants and wild types; transcriptome sequence alignment of PIF4 of strong and weak pods genotypes for possible SNPs difference; sequencing of PIF4 gene of weak and strong pods genotypes and biochemical analysis through tissue fixation of pods. The role of PIF4 in pod shattering has been confirmed through RIT of mutant and wild type Arabidopsis. The SNP difference found in the A3 copy of PIF4 may be used in marker assisted selection (MAS) for pod shattering in Brassica napus. In biochemical analysis, genotypes with strong pods have more lignin in their pods as compared to those having weak pods.