Role of Pakistan as a Frontline State and its Sociocultural Impact

Location of Pakistan has great geo-political and geo-strategic significance in the South Asian region. Pakistan played an important role for the strategic goals of the United States during the ruling periods of President Zia-ul-Haq and President Musharraf. As a frontline state Pakistan received economic and military assistance, during the period of President Zia-ul-Haq, for supporting US interests against Soviet invasion in Afghanistan. After 9/11, due to its strategic location, Pakistan became the ally of the United States against terrorism and the biggest beneficiary of US economic assistance in the region. However, as a result of these policies Islamic extremism, sectarian violence and ethnic cleavages cropped up in the society, shattering social and cultural values. The purpose of this research paper is to highlight the key role played by Pakistan as a frontline state and its impact on the Pakistani society. This research paper follows descriptive and analytical methods.

Approximation of Oscillatory Integrals and Integral Equations Via Localized Kernel Based Method

One of the most important area of mathematical sciences is the numerical evaluation of oscillatory integrals and integral equations. Variety of applications constitute these integrals including acoustic scattering problems. In the present research we construct numerical schemes which are based on localized kernel based method for approximation of oscillatory integrals and integral equations. This localized setting results in small size interpolation matrices. The advantages of these small size interpolation matrices would enable us to cope with the huge amount of data points and solve the integral problems for thousand of scattered points in complex shaped domains. In the first part of this thesis the problem of integration is transformed into ODE’s and PDE’s and apply a localized collocation scheme to solve the resultant ODE’s and PDE’s numerically. For large amount of data points in the integration domain, the present numerical scheme has the ability to compute the oscillatory integrals. In the second part of this thesis we extend our work for computing more generalized fast oscillatory integrals. The method is efficient for computing more generalized oscillatory integrals involving Bessel functions with large oscillation parameter. In the third part we consider finite highly oscillatory Hankel transform arises in electromagnetic and acoustic scattering problems, a numerical scheme is developed which is based on the local kernel based method. In the last part of this thesis we used radial kernels for computing more generalized fast oscillatory integral equations. The present method is efficient for computing oscillatory integral equations with large oscillation parameters.