امام بخاریؒ کا صحیح بخاری میں مؤطا امام مالک پراعتماد؛ تجزياتى مطالعہ

In every period, hadith experts have contributed to the preservation of the Hadith. Imam Malik composed M’ūaṭa to safeguard the hadith, and Imam al-Bukhari afterwards turned to al-Mu'tah for assistance. The research methodology of this paper is an analytical study of the reliance of Imām Bukhārī on Al-M’ūaṭa Imam Mālik in Ṣaḥiḥ Bukhārī. In our article, we came to the conclusion that Imam Bukhari recounted a total of six hundred sixty-eight narrations from Imam Malik in his Sahih, of which six hundred four are scriptural narrations that he took from ten versions of Al- M’ūaṭa. As a result, these narrations make up more than ninety percent of all narrations. Contrasted with in his Sahih, Imam Muslim narrates three hundred eighty-nine narrations from Imam Malik, three hundred twelve of which are scriptural narrations that he has derived from eight of narrations of al M’ūaṭa's. These narrations so account for more than eighty percent of all narrations.

Nonlinear Wave Propagation in Photonic Band Gap Materials

This research work is concerned with the nonlinear wave propagation in photonic band gap (PBG) materials. Photonic band gap materials are artificial periodic structures that exhibit bands and gaps for the propagation of light in certain frequency ranges depending on the periodicity of the structure. Nonlinear PBG materials can be defined as the periodic structures that have the ability to control light with light and produce a structural enhancement of nonlinear effects. The optical properties of such materials can show dramatic changes with the changing intensity of light. An interesting manifestation of nonlinear effects in these structures is the formation of gap solitons and the associated phenomenon of optical bistability. Promising applications for such phenomena include optical switches, logic gates, transistors etc. Recent theoretical and experimental studies have suggested the possibility of creating artificial metamaterials that can possess a negative real part of magnetic permeability and/or a negative real part of electric permeability and are known as double and single negative metamaterials (DNG, SNG). The main focus of the present study is to explore the new features of the nonlinear wave propagation due to the inclusion of double negative and single negative metamaterials in PBG structures. The conventional PBG structures are characterized by Bragg gaps whereas in PBG structures containing DNG and SNG metamaterials, mechanisms other than Bragg reflections can produce gaps such as the zero-n and the zero- φ eff gaps. The real advantage of these new gaps is the fact that these are relatively insensitive to the incident angle and disorders in the structures. iiIn the present work we have considered only one-dimensional structures. The transfer matrix approach and the characteristic matrix approaches are mainly used to study the wave propagation through these structures. We have also applied the Kronig-Penney delta function approach in one of our investigations. The electric field profiles for the zero-n gap soliton, zero-φ eff gap soliton and Bragg gap soliton are plotted and a comparison of their characteristics is made. Similarly the optical bistability curves for the one-dimensional structure containing alternate DNG and regular materials and containing SNG materials are also plotted. We have considered normal as well as oblique incidence. The properties of an angular gap in one-dimensional structures containing SNG layers are investigated. We have also investigated some defect structures. The properties of the resulting localized mode are studied. The results of the investigations are summarized at the end.