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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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شیخ ظہور الحسن

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افسوس ہے کہ شیخ ظہورالحسن صاحب سابق ریوینیو سکریٹری حکومت اتر پردیش نے کراچی میں انتقال کیا، وہ اپنے محکمہ کے ماہر اور اس صوبے کے لائق ترین عہدہ داروں میں تھے، حکومت میں ان کا بڑا وقار تھا، عملاً بڑے مذہبی اور دیندار تھے، مذہبی مطالعہ وسیع تھا، دینی کاموں سے بڑی دلچسپی تھی، ترک وطن سے پہلے دارالمصنفین کی مجلس انتظامیہ کے رکن تھے، کئی سال ہوئے ریٹائرڈ ہوچکے تھے، ان کے لڑکے پاکستان میں بڑے عہدوں پر تھے، لیکن وہ خود ہندوستان ہی میں مقیم رہے، ۱۹۶۵؁ء کی جنگ سے کچھ پہلے لڑکوں سے ملنے کراچی گئے تھے، اسی دوران میں جنگ چھڑگئی، اس لیے لڑکوں کے اصرار سے وہاں مستقل قیام اختیار کرلیا، اور وہیں قلبی دورے میں وفات پائی، مسلمان عہدہ داروں میں ایسے لائق اور دیندار کم ہوتے ہیں، اﷲ تعالیٰ اس مرد مومن کی مغفرت فرمائے۔ (شاہ معین الدین ندوی، جنوری ۱۹۷۲ء)

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Study of Soft X-Rays from Depleted Uranium Assisted Pre-Ionized Plasma in a Low Energy Mather Type Plasma Focus Device

This thesis describes the experimental investigations of the soft X-ray emission and the plasma dynamics in a Mather type plasma focus device operated in different gases (Neon, Helium, Deuterium, Neon+Helium) by changing different parameters like charging voltage, capacitor bank energy, discharge current, anode shape, gas pressure and influence of preionization. The objectives of the studies are to enhance soft X-ray emission. The Rogowski coil, high voltage probe, PIN diodes and pinhole camera masked with appropriate absorption filters are used for diagnostics. Specifically attention is paid to tailor the radiation in different energy windows i.e. 1− 1.56 keV and 4−8.33 keV. A single 12.5 μF capacitor charged at 19-23 kV (2.25-3.3 kJ), with a field distortion type sparkgap as switch is used to transfer the energy to the system. In the first experiment, the effect of preionization caused by depleted uranium (92U238) on soft X-ray and total X-ray emission is investigated by varying the neon gas pressure with and without preionization. The X-ray yield and pinhole images are found strongly influenced by the preionization and the pressure of the working gas. The soft X-ray yield of 82.5 ± 4.0 J in 1.3–1.56 keV energy window is obtained with preionization of neon gas at the optimum pressure, leading to the conversion efficiency of 2.5% of the stored energy. The total X-ray yield with preionization is 195 ± 9.0 J in 4π geometry measured at the optimum pressure giving conversion efficiency of 5.9%. The time integrated images indicate the broadening of X-ray emission zone with preionization. The second experiment is performed with helium as filling gas. The highest soft X-ray yield of 0.25 ± 0.01 J is obtained without preionization, which increases to 0.50 ± 0.02 J with preionization at optimum pressure. The total X-ray yield 1.50 ± 0.07 J without preionization, is enhanced to 2.44 ± 0.11 J with preionization. The plasma focus duration has been increased with preionization conforming enhancement and improvement in X-ray. This has also been conformed by the symmetric and enlarges pinch images with preionization.