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Optimal Couples of Rearrangement Invariant Spaces for the Riesz, Bessel and Fractional Maximal Operators

Thesis Info

Access Option

External Link

Author

Nazeer, Waqas

Supervisor

Georgi Eremiev Karadzhov

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/123456789/1575

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726812845

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We prove continuity of the Riesz, Bessel and Fractional Maximal Operators in optimal couples of rearrangement invariant function spaces defined in Rn with the Lebesgue measure.
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عروجِ آدم خاکی سے یہ انجم سہمے جاتے ہیں

عروج آدم ِخاکی سے ا نجم سہمے جاتے ہیں
نحمدہ ونصلی علی رسولہ الکریم امّا بعد فاعوذ بااللہ من الشیطن الرجیم
بسم اللہ الرحمن الرحیم
معزز اسا تذہ کرام اور میرے ہم مکتب شاہینو!آج مجھے جس موضوع پر اظہار خیال کرن ہے وہ ہے:’’عروج آدم خاکی سے انجم سہمے جاتے ہیں‘‘
جنابِ صدر!
انسان کو اللہ تعالیٰ نے اعلیٰ مقام عطا فرمایا ہے، اس کو کلامِ مجید میں اشرف المخلوقات کا تاج پہنایا ہے، اس کی عظمت کو فرشتوں سے منوایا ہے، اس کو مسجودِ ملائکہ بنایا ہے، اس کے شرف سے دیگر مخلوقات کو آگاہ فرمایا ہے۔
فرشتے سے بہتر ہے انسان بننا
مگر اس میں پڑتی ہے محنت زیادہ
صدرِذی وقار!
انسان جب اپنی تخلیق کا مقصد پہچان لے، اپنے وجود کے بارے میں آگاہی حاصل کر لے، اپنی حیات کوصحیح خطوط پر گزارنے کا سلیقہ حاصل کرلے، اپنے لمحات زیست سودمند مواقع کی تلاش میں صرف کر دے، اپنے حواسِ خمسہ کا صحیح استعمال کرنا سیکھ لے تو اس کو اپنی منزل آسمانوں پر نظر آنی شروع ہو جاتی ہے۔
عقابی روح جب بیدار ہوتی ہے جوانوں میں
نظر آتی ہے ان کو اپنی منزل آسمانوں میں
صدرِمحترم!
اس آدم خاکی نے جب سرحدوں کا رخ کیا تو دشمن لرزہ براندام ہو گئے ، دشمن کو قہر آلود نظروں سے دیکھا تو ان کے رونگٹے کھڑے ہو گئے ، اعداء پر ضربِ کلیمی لگائی تو فضاء گونج اُٹھی، اس نے دشمنوں کی زندگی اجیرن کر دی ، اس کی نسلوں تک مار کرنے والے تصورات پیش کیے، دشمنِ اسلام کو بتایا کہ خدا اور اس کے رسولﷺ کے پیروکارکبھی گھبراتے نہیں۔
دو نیم ان کی ٹھوکر سے صحرا و دریا
سمٹ کر پہاڑ ان کی ہیبت سے رائی
جنابِ صدر!
انسان جب کھیتوں ، کھلیانوں میں پہنچتا ہے تو انہیں...

عصر حاضر میں آرائش انسانی کے جدید سائنسی انقلابات اور اسلامی نقطۂ نظر

Every single thing made by Allah, the Creator of all things has its own essence of beauty and attraction to it. However, out of all the stunning creations of Almighty Allah, humans are the most superior creation which has the ability to gain knowledge. We, humans, have been given the power by Allah to explore the rest of his creations in nature and fully understand the beauty and functions of each and every aspect of it in order to take advantage of it. Islam is a religion based on nature and Shariah is a law which lays down rules for Muslims to follow which also allows us to move forward with the advancements in the world. It has a unique way of life for any person of any time to follow which cannot be found in any other religion. However, any development which takes place creates new problems and new obstacles which can only be manoeuvred by the researchers and scholars of that specific time who will decide the right use of that commodity and explain the commodity. This will allow the new invention to be utilised fully. Moreover, in this world many people want to look their best and feel their best and will be willing to go through many ways for their ideal look. With this desire, many jobs become associated with achieving this for people. Nonetheless, this leads to people putting their time, money and effort into something which can risk their entire life which can result in either an advantage or disadvantage. Moreover, they should know whether this is against their religious values. Those acts which cross your religious boundaries and are considered “haram”or not permitted are wrong. Thus, it is necessary for us to first consider whether it is crossing the boundaries and proving to be disadvantageous for you so that the wrong norms and values are not transmitted into the future generations.

Effects of 1-Trifluoromethoxyphenyl-3- 1-Propionylpiperidin-4-Yl Urea on Cardiovascular and Metabolic Dysfunctions in Obese and Diabetic Rodent Models

Obesity and diabetes are increasingly important public health problems across the world.Obesity is associated with higher risk for chronic diseases including cardiovascular disease and type 2 diabetes. The increasing trend of diabetes and obesity is imposing a huge burden on health-care systems. Despite therapeutic management with conventional agents obesity and related cardiovascular and metabolic complications remain the major health concerns around the globe. The currently available drugs for obesity and diabetes are associated with multiple adverse effects. Therefore there is an urgent need to develop newer drugs, which are safe and effective. Soluble epoxide hydrolase (sEH) inhibitors offer a novel promising therapeutic target to manage diabetes, obesity and associated complications. Inhibition of soluble epoxide hydrolase increases epoxyeicosatrienoic acids (EETs) levels. EETs are epoxide derivatives formed from arachidonic acid by the action of cytochrome P450 (CYP) epoxygenases specially theCYP2C and CYP2J subfamilies thatmetabolize arachidconic acid to various EETs, such as 5,6-, 8,9-, 11,12-, and 14,15-EETs. EETs are important regulators of the cardiovascular system and have been shown to exert potent vasodilators, anti-inflammatory, antithrombotic, cardio protective and antioxidant effects. Growing number of evidences suggest the over activity of sEH in obesity and diabetes which leads to functional deficit of EETs in the body. Decreased level of EETs is implicated in pathophysiological mechanisms of obesity and diabetes associated complications. Stabilization of EETs through pharmacological inhibition of sEH or sEH deletion has been shown to have beneficial effects in various conditions including pain, inflammation, hypertension and ischemia. Based on diverse physiological role of EETs, the current project was aimed at evaluation of 1-trifluoromethoxyphenyl-3-(1-propionylpiperidine-4-yl) urea (TPPU), a novel soluble epoxide hydrolase inhibitor (sEHI) against fructose induced diabetes and high fat diet–induced obesity in related cardiovascular and metabolic complicationsin rats. TPPUis the most potent soluble epoxide hydrolase inhibitor, metabolically stable and having good pharmacokinetic profile. Although the potential of soluble epoxide hydrolase inhibitor for the treatment of obesity related cardiovascular diseases and metabolic dysfunctions appear promising, it has not been fully established. The current study revealed that rats administered with fructose solution for 12 weeks developed classic symptoms of metabolic syndrome. Controlled rats fed on fructose showed tremendous increase in blood pressure, blood glucose and insulin levels as well as levels of TGs, cholesterol, LDLs were increased (P<0.001). There was significant increase of ALT, ALP, urea and uric acid levels in rats fed on fructose. Endothelium dysfunction was observed in fructose fed animal with attenuated response to acetylcholine.Animals that were fed on fructose and treated with TPPU (2 mg/kg p.o) for 12 weeks improved insulin resistance , decreased blood pressure, blood glucose levels and lowered serum cholesterol, TGs and LDL while HDL levels was elevated, compared to untreated control animals. The fructose induced endothelium dysfunction was also improved with TPPU treatement. The histopathology and immunohistochemistry results of the TPPU treated animals exhibited regenerative effect on ?-cells of the pancreas in diabetic rats. There was increase in the number of islet of langerhans in the pancreas of diabetic rats treated with TPPU compared with untreated rate. The Immunohistochemical staining with anti-insulin antibody confirmed a marked reduction in insulin secreting cells in diabetic untreated rats while TPPU treated group showed normal mass of secreting cells. The results of the present study revealed regeneration in the pancreatic ?-cells in diabetic rats treated with the TPPU (2 mg/kg p.o) for 12 weeks.The present work showed a significant increase in levels of serum liver enzymes (ALP, ALT) in the fructose-fed untreated animals which were significantly decreased in TPPU treated animals. The microscopic examination of the liver sections of diabetic untreated group exhibited great improvement of cellular architecture in the TPPU treated animals. In the present study, obesity was induced rats by feeding them on HFD (Cholesterol 40%, carbohydartes 40% and proteins 20%) for two months. Rats fed on HFD had considerable increase in their body weights compared to age-matched control animals fed on regular diet (Normal control). The increase in body weights of rats fed on HFD and simultaneously administered with TPPU (2 mg/kg p.o) for the two months was significantly decreased (P<0.01). There was also significant (P<0.01) increase in the systolic blood pressure of animals and vascular dysfunction with decreased relaxant response of vessels to acetylcholine and sodium nitroprussidein animals fed on HFD compared to control rats fed on regular diet. This increase in systolic blood pressure and vascular dysfunction was attenuated by TPPU treatment in animals fed on HFD. , In the present study, HDF caused significant increase in plasma creatinine levels (P<0.001) in animals compared to those fed on normal diet. The increased in plasma creatinine levels were inhibited in animals fed on HFD and treated with TPPU for two months. HFD-induced obesity is most commonly associated with systemic inflammation. The present study also investigated systemic inflammation by measurement of circulating obesity markers. The adiponectin plasma levels were highly decreased (P<0.001) in rats fed on HFD for two months compared to control rate fed on normal diet (p < 0.001).Adiponectin levels were increased in rats fed on HFD and treated with TPPU. Leptin resistance was also increased in obese animals associated with higher leptin levels inanimals fed on HFD compared to those fed on normal diet. The increase in leptin levels was significantly decreased in animals fed on HFD and treated with TPPU for two months.In the present investigation obese animals had increased levels of plasma insulin compared to the control animals fed on regular diet (normal control). This increase in insulin levels was significantly suppressed (P< 0.001) by TPPU treatment in rats fed on HFD. The other metabolic parameters such as plasma cholesterol, TGs, LDLs were increased and HDL decreased in obese animals fed on HFD compared to those fed a regular diet.HFD also caused tremendous increase in serum ALT and ALP. The increase in lipid levels was inhibited and HDL levels enhanced in animals fed on HFD and treated with TPPU. TPPU treatment also reversed HFD induced ALT and ALP abnormalities in rats caused by HFD treatment. The decrease in liver enzymes was also reflected in the finding from microscopic examination of the liver sections showing marked improvement in the liver architecture in animals fed on HFD and treated with TPPU compared to those treated with vehicle and fed on HFD. These findings indicate that TPPU could be potential new drug to manage obesity induced cardiovascular and metabolic complications. These findings indicate that therapeutic and/or preventive potentials of TPPU against fructose induced diabetes and HFD-induced obesity and the related cardiovascular and metabolic complications in rats.