It is well known that the theory of inequalities is considered as one of the central areas of mathematical analysis. It has many important applications in numerous scientific fields. In recent years, considerable attention has been given to this field in order to find the generalizations, variations and applications of different inequalities. The aim of this thesis is to prove several inequalities involving some special functions in terms of a new parameter k > 0. We can call these functions as special k-functions. Here, we do work on k-analogue gamma, beta and psi functions. This research work consists of eight chapters. In first chapter, we prove the inequalities involving k and q, k-analogue of gamma and psi functions. In chapter 2, We derive some classical inequalities of Chebyshev, H¨older and Gr¨uss type involving gamma and beta k-functions. In chapter 3, we discuss some basic properties, recurrence relation and special cases of incomplete gamma and beta functions in terms of the parameter k > 0. Some inequalities involving the incomplete beta k-functions are also given. In chapter 4, we prove some Gr¨uss type integral inequalities involving the generalized RiemannLiouville fractional integral in terms of parameter k > 0. In Chapter 5, the Ostrowski type inequalities involving the left and right-sided Riemann-Liouville fractional integrals are established in terms of the parameter k > 0. From our results, the classical Ostrowski inequalities can be deduced as some special cases. In chapter 6, we introduce the k-analogue of Hadamard fractional integral with some properties. We prove different types of inequalities involving this newly defined k-fractional integral. In chapter 7, we define the k-deformation of the fractional integral of a function with respect to another function and the inequalities involving the newly defined k-fractional integrals are also be proved. In last chapter, we introduce some properties of beta k-distribution and present some inequalities involving beta k-distribution via some classical inequalities, like Chebyshev’s inequality for synchronous (asynchronous) mappings and H¨older’s inequality. Also, we discuss the inequalities for harmonic mean, variance and coefficient of variation of βk random variable involving the parameter k > 0. Finally, we give conclusion of the present study and recommendations for the future work. Additionally, published work of the author has also been attached at the end of the thesis.
تلوک چند محرومؔ مشہور استاد سخن تلوک چند مرحوم کی وفات ادبی دنیا کا افسوس ناک سانحہ ہے، وہ ہماری پرانی بزم ادب کی یادگار تھے، اردو شاعری کے اساتذہ میں ان کا نہایت ممتاز مقام تھا، فن شاعری پر ان کی نظر بڑی گہری اور وسیع تھی، اور ان کا کلام جدت و قدامت کے صالح عناصر کا سنگم تھا، ان کو ہر صنف شاعری پر یکساں قدرت حاصل تھی، ان میں پرانی تہذیب کی بہت سی خوبیاں اور وضع داریاں جمع تھیں، ان کا دل دوسرے مذاہب اور اس کے بزرگوں کے لئے بھی وسیع تھا، بارگاہ نبویؐ میں بھی ہدیۂ عقیدت پیش کرتے تھے، قوم و ملک نے بھی ان کے کمالات کی پوری قدر دانی کی اور ان کو ہر طرح کے اعزاز و اکرام سے نوازا، ان کی پوری زندگی علم و ادب کی خدمت میں گزری، اپنے بعد متعدد ادبی یادگاریں چھوڑ گئے، ان کی مادی یادگار مشہور شاعر جگن ناتھ آزاد ہیں، جنھوں نے اپنے محترم والد کی بہت سی خوبیاں حصہ میں پائی ہیں۔ (شاہ معین الدین ندوی، فروری ۱۹۶۶ء)
Human nature has become collectivist. And man's survival is in living together and fulfilling the necessities of life through mutual cooperation. As a result, a system of mutual relation is formed; the state has relation with the other state just as the individual has relations with the individual. If there is no balance and harmony in this relationship, then mutual conflict and tension is natural. In fact, international law has its beginnings from the Islamic era. The Prophet (peace and blessings of Allah be upon him) conveyed the universal message of the Islamic State throughout the world through his foreign Strategy and diplomatic letters. Away from the concept of globalization, the importance of foreign relations has become more important in modern times. Therefore, just as it is necessary for the Islamic state to harmonize with the world conditions and events according to the requirements of the modern era, it is also necessary for the Islamic state to keep its ideological position in mind in all situations. So that the foreign strategy of the Islamic State is established on strong foundations and its principles are formulated in the light of International Law and Sharia.
Keywords: International Law, Foreign Policy, Foundations, Islamic era
Sewage sludge is being heaped in the overcrowded urban cities and its safe disposal to agriculture soils is becoming a serious concern around the world. In developing countries like Pakistan, the use of sewage sludge is becoming common due to its free of cost and continuous availability. Most of the sludges contain large amounts of adsorbed and precipitated forms of heavy metals which could become a serious environmental threat. Among these toxic metals, cadmium (Cd) is grabbing much attention due to its high solubility and subsequent biosphere toxicity. The present studies aim at decreasing Cd bio-availability and producing the safe food from normal soils using sewage sludge. The effect of ageing and temperature on the release of Cd from the sewage sludge was monitored in an incubation study. The study was carried out at different temperatures (22 and 45 ºC) to monitor Cd availability and organic matter degradation in soils receiving sewage sludge @ 0, 15 and 30 g kg-1. Sludge addition and high temperature made obvious changes in fulvic acid FTIR spectra mainly at 1050-1250 cm-1 region. While humic acid spectra showed more aromatic character which are chemically and biologically more stable for Cd retention. Sequential data revealed significant decrease in organic matter bound Cd and increase in exchangeable Cd at high temperature (45 ºC). High temperature also increased the proportion of residual Cd and decreased the reducible Cd fraction with ageing, indicating high temperature had major influence on redistribution of Cd forms. It is concluded that long term monitoring is required for estimating the availability of these immobilized and transformed Cd fractions and organic matter constituents after sewage sludge application in hot and dry climate of arid and semiarid regions. In the subsequent incubation study, influence of organic matter derived humic and fulvic acids were investigated on Cd availability at various levels of soil pH and moisture. Cadmium transformations were investigated in soils (pH 7.10, 8.48, 9.52) treated with sewage sludge @ 0, 15 and 30 g kg-1 and maintained at 50 and 70% moisture soil WHC. Results revealed that high pH soils (8.48 and 9.52) recorded more changes in humic and fulvic acids FTIR spectra. Fulvic acid demonstrated that its activity depends on labile alcoholic and carboxlic groups. While, humic acid exhibited more humification and aromatic character with ageing. Sequential extraction data showed decreased Cd availability in high pH soils by decreasing the proportion of available Cd but low pH soil tended to favor Cd availability. These results suggest that maintaining soil moisture at 50% WHC could transform Cd more into non-available fractions due to increased Cd precipitation with soil colloids and by forming humate complexes. In this preliminary pot study, wheat cultivars (Shafaq-06, AARI-11, Ufaq-06, Fareed-06, Sehar-06, Aass-11, Gomal-08, Punjab-11, Millat-11, Galaxy-11, Uqab-00 and NARC-11) were planted in a sand medium contaminated with Cd @ 0, 5, 10 and 15 mg L-1) to screen out the Cd tolerant cultivars. The NARC-11 seedlings performed better against different Cd stress levels in terms of germination parameters (rate, energy and index) and root and shoot length. The Galaxy-11 was found to be more tolerant than other cultivars in terms of tolerance index at higher levels of Cd stress (15 mg Cd L-1). The Shafaq-06 emerged as non-tolerant and didn’t perform well in terms of above mentioned parameters. From these results, it is evident that NARC-11cultivar being Cd tolerant could be grown on Cd contaminated soils but it demands further studies to testify its tolerance under natural field conditions. Later a pot study was designed to assess the effects of Cd release and humic and fulvic acids transformations on wheat growth in sewage sludge amended soils. NARC-11 and Shafaq-06 were sown in soil amended with sewage sludge (0, 15 and 30 g kg-1 soil). Application of sewage sludge resulted in better growth and Cd concentrations in wheat grains of both cultivars remained within permissible limit. Spectroscopic analysis revealed that fulvic acid underwent more spectral changes and humic acid showed more progressive maturity due to its increased hydrophobic aromatic groups. Sequential extraction data indicated a significant decrease in mobile fractions of Cd after crop harvest. It is concluded that sewage sludge application could be useful for the improvement of wheat production due to the formation of stable humate complexes and resulting in decreased Cd availability. Another pot study was set to determine the effect of humic and fulvic acids transformations on wheat growth (NARC-11 and Shafaq-06) in naturally Cd contaminated soils (1.25, 2.29, 3.23 and 4.18 mg kg-1). Results revealed that both the cultivars showed better growth in soils with Cd concentrations less than 3 mg kg-1 soil. Cadmium concentrations in wheat grains of both the cultivars grown in Cd contaminated soils (Cd>3 mg kg-1) were found beyond the permissible limits. After wheat harvest, minor percent changes in different fractions of Cd indicated that Cd has been stabilized in soil due to long term irrigation with raw city wastewater. The FTIR analysis also depicted minor changes in humic and fulvic acid spectra, due to humification organic matter from more stable humic complexes.