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Two–Weight Criteria for Potentials With Product Kernels on the Cone of Non-Increasing Functions

Thesis Info

Access Option

External Link

Author

Murtaza, Ghulam

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/bitstream/123456789/2760/1/1469S.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727635989

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Necessary and sufficient conditions governing one and two weight inequalities for one-sided strong fractional maximal operators, one-sided and Riesz potentials with product kernels are established on the cone of non-increasing functions. From the two– weight results it follows criteria for the trace inequality Lp (Rn ) → Lq (v, Rn ) bound- + + dec edness for these operators, where v, in general, is not product of one-dimensional weights. Various type of two-weight necessary and sufficient conditions for the dis- crete Riemann–Liouville transform with product kernels are also established. The most of the derived two-weight results (continuous and discrete) are new even for potentials with single kernels
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ایک تو ہے صدمات نے گھیرا

ایک تو ہے صدمات نے گھیرا
اس پہ اپنی ذات نے گھیرا

کیسے ہوش سنبھالوں اپنے
غم کی ہے بہتات نے گھیرا

ہم کو یاد تھے ہم ہی آئے
پیروں کو جب پات نے گھیرا

اپنی ذات نفی میں ڈالی
ہم کو جب اثبات نے گھیرا

اک مفلس کی شامت آئی
کچا گھر برسات نے گھیرا

اس نے کبھی پھر دن نہیں دیکھا
جس کو ہجر کی رات نے گھیرا

المنھج اللغوي في التفسير و تاريخه

The Holy Quran was revealed in Arabic Language, it is, therefore necessary to seek Arabic Diction to gain the direct guidance from it. The companions of Holy Prophetr, Tabeen, and the reverent Imams strictly rebuked those interpreters who interpret the Holy Quran without having command over Arabic Language. The verses of Quran that are clear in comprehension, explicit and easy, do require the source of interpretation as “Arabic Diction”. This method highlights the positive trends to Arabic Diction. But in the matter of ambiguity and resemblance in verses and deduction of Masael, this Diction will be given second priority. Mere Diction and Arabic Socio-Diction may not be titled as most authentic. Diction is not the ‘last word. ’ The very first priority will be given to the verses of Quran, Hadith e Nabvi and Quotations of Companions of Holy Prophetr. The companions themselves were the native Arabs but they used to do consult some Quranic terms with the Holy Prophetr. As time passed, some strayed sects and atheists ignored this positive trend (Tafseer-bil-Mathur), and accustomed a new trend of interpretation of Holy Quran i. E. Depending upon Arabic Diction only so that they may endorse their own thoughts. It was a negative source of interpreting the Holy Quran i. E. Only by Arabic Diction. The present article explores its historical perspectives after evaluating its negative trends. The Motazila sect got this trend nourished. The representing interpretations of Holy Quran of this trend have been analyzed in this article. At the end, Molana Ameen Ahsan Islahi’s approach to Diction and his Tafseer ‘Tadabbur e Quran’ has been evaluated.

Anthraquinone Sulfonamides and Derived Heterocycles: Design, Synthesis, Biological Evaluation and Computational Studies

Leishmaniasis, a worldwide prevalent disease, is still enjoying the ruling with no proper medication; and to add to this current gloomy scenario the disease causing parasite Leishmania is becoming resistant to the ongoing medication that is being practiced now a days. Hence, the need is to search for reasonable, safe and targeted drugs; the present research is one such effort in this direction. To begin with, Leishmanolysin (GP63), zinc metalloprotease, expressed over the surface of Leishmania species was selected as drug target due to its virulence and reason for parasite resistance. A library of benzimidazole derivatives (1-37) was synthesized and screened for its antileishmanial potential against L. major. All the compounds were found potent antileishmanial with IC50 values in the range of 0.62-0.92 μg/mL as compared to amphotericin B (standard drug) IC50 value 0.56 μg/mL. 2-(Thiophen-2-yl)-1H- benzimidazole (19) and 2-(1H-indol-3-yl)-5-nitro-1H-benzimidazole (34) were identified as the lead compounds of the library with IC50 value of 0.62 μg/mL. ADMET properties of the entire library were also predicted by using ADMET PredictorTM and were observed to be safe. Molecular docking studies carried out on all the members of library and amphotericin B by using MOE software, indicated that the most active compounds fitted at the centre of binding pocket of GP63 built by amino acid residue His264, His268, His334 and Zn578. On the basis of molecular docking results, receptor based pharmacophore model was built containing three Aro|Hyd features and one Acc&ML feature. This pharmacophore model was used to design new scaffolds for antileishmanial compounds. Four libraries, 2-(2- aryl/heteroarylbenzimidazol-1-sulfonyl)anthraquinones (38-69), N-(heteroaryl)-anthraquinon-2- sulfonamides (70-95), aryl anthraquinon-2-sulfonates (96-111) and N-(anthraquinon-2-sulfonyl)-amino acid methylesters (112-123) were designed and all the cmpounds were found as hit by pharamocophoric search. Their antleishmanial activities were predicted by QSAR model; built by MOE software by selection of 94 descriptors and partial least square (PLS) method on experimental antileishmanial activity of 37-mebered library and amphotericin B, validated by internal and exernal test sets with correlation coefficient (R2) 0.7762. All the compounds belonging to four libraries (38-69, 70-95, 96-111 and 112- 123) were found potent antileihmanial with predicted activity in the range of 0.5435-0.9940. All the compouds were observed safe according to predicted ADMET properties and Lipinski’s rule of five (Ro5). Later, these four designed libraries were synthesized and characterized by physical constants and spectroscopic techniques for onward screening for their antileishmanial potential against L. major by using amphotericin B as standard control which confirmed that all the compounds were potent antileishmanial. Compliance of the predicted activity by QSAR model with observed activity from in vitro antileishmanial activity resulted in identification of the same lead compounds in each library 38-69, 70-95, 96-111 and 112-123 i.e. 2-(5-Nitro-4-methoxyphenyl-1H-benzimidazol-1-sulfonyl)anthraquinone (61) (predicted activity 0.6794, IC50 0.67 μg/mL), 2-(1H-benzo-1,2,3-triazol-1-sulfonyl)anthraquinone (91) (predicted activity 0.5579, IC50 0.57 μg/mL), 2-(1H-pyrazol-1-sulfonyl)anthraquinone (90) (predicted activity 0.5435, IC50 0.58 μg/mL), benzyl anthraquinon-2-sulfonate (100) (predicted activity 0.7615, IC50 0.76 μg/mL) and N-(anthraquinon-2-sulfonyl)-2-phenylglycine methylester (123) (predicted activity 0.7305, IC50 0.75 μg/mL). Pharmacophore based molecular docking studies carried out on all the eighty six compounds on GP63 by MOE software showed hydrophobic interactions, hydrogen bonding and metal ligation interactions with His268, His264, His334 and Zn578, respectively. This entire set of experiments in both dry and wet labs led to a successful designing of a variety of anthraquinon-2- sulfonamides as a novel scaffold having strong antileishmanial effect.