In the present study, inspired by the studies in the literature, we consider Min matrix and its Hadamard exponential matrix family whose elements are Chebyshev polynomials of the first kind. Afterwards, we examine their various linear algebraic properties and obtain some inequalities. Furthermore, we shed light on the results we obtained to boost the clarity of our paper with the illustrative examples. In addition to all these, we give two MATLAB-R2023a codes that compute the Min matrix and the Hadamard exponential (...) matrix with Chebyshev polynomials of the first kind entries, as well as calculate some matrix norms. (shrink)

In addition to the sawms that are ordered and recommended in Islam, there are also sawms that are prohibited or not welcomed in certain times and situations. Sawm on the thirtieth day (yawm al-shakk) of the month of Shaʿbān, when the month of Ramaḍān cannot be decreed to have started, is one of them. The Prophet mentioned that Allah allocated the hilāls for people to arrange the times and in principle, he wanted to start fasting with the sighting of the (...) moon, and to calculate the amount of the month in case of cloudy weather in order to avoid any doubt. Despite this, different narrations have emerged in the ḥadīth literature about fasting on the yawm al-shakk and this situation has even been reflected in the sectarian jurisprudence. As a result, three basic views and practices emerged during the time of ṣaḥabah, tābiʿūn and tebeʿu’t-tābiʿīn that considered it makruh to fast except for the usual fast that coincided with the day in question, that fasting could be kept, and that the decision of the administration should be taken into account in order to prevent possible conflicts. In this article, the riwāyahs about fasting in the yawm al-shakk pertaining to the Prophet, the ṣaḥābah, tābiʿīn and the later period are discussed and analyzed with their practical aspects, and finally, the views of the sect imams are also included. (shrink)

Ḥadīths have been preserved and recorded in various ways since the Companions. These activities continued dur-ing the Tābiīn (the successors of the Companions) Period. So much so that these methods have formed the infra-structure of other methods that will emerge later. In this context, before the 70's (A.H.), works named al-Aṭrāf appeared. However, these first works consisted of the notes that they wrote some of the ḥadīths before coming to the science assemblies to help students remember in ḥadīth learning. Ḥadīth (...) scholars have adopted and applied this method in a very short time. Thus, this method also provided the opportunity for the formation of the first written ḥadīth documents. In addition, with the writing of ḥadīth works, it has turned into a system of al-Taṣnīf and al-Takhrīj in order to make maximum use of these books and to reach the source of a narration that is tried to be found. As a result, al-Aṭrāf literature, which can be evaluated as index, has emerged. As a matter of fact, the first examples of this literature, arranged alphabetically according to the names of the Companions or ḥadīth texts, started to be given since the 4th (A.H.) century. In this way, the way to quickly benefit from the accumulation of the ḥadīth area has been opened. This article focuses on the historical adventure of al-Aṭrāf and their contributions to the ḥadīth field, as well as literature knowledge of the works of al-Aṭrāf which are of great importance for both al-Kitābah, al-Taṣnīf and al-Takhrīj.Summary: In order to understand and practice the religion of Islām, great efforts have been made both in the past and later periods in order to convey the words, verbs, interpretations and qualifications of the Prophet to the next genera-tions as they are protected from all kinds of falsification and changes. This process, which started with memoriza-tion and partly writing at the beginning, has gained momentum with the use of other methods since the period of Tābiīn. In this context, the method of aṭrāf was started to be used to facilitate memorization, to be a reminder, and to transmit the ḥadīth through the hearing of the ḥadīth sheikh. The first examples of this method were seen by Muḥammad b. Sīrīn during his participation of his teacher Abīde/Ubayda Salmān in the scholarly assembly. At this stage, the relevant method was manifested as writing a passage from the beginning of the ḥadīth. Thus, the aforementioned method, which had great contributions in recording and memorizing ḥadīths in accordance with the original, has gained widespreadness and application among the ḥadīth scholars like Ubeydullaḥ b. Omar, Shu’bah b. al-Ḥajjāj, Sufyān as-Sawrī, Mālik b. Anas, Ismāīl b. Ayyāş and Vakī b. Jarrāḥ in a very short time. In this context, Yazīd b. Zuray’ stated that he went to his teacher without his aṭrāf work of ḥadīths which he transmitted from Shu’bah and listened to ḥadīth in this way. Therefore, this mentioned event is important in terms of putting forth that Shu’bah allowed the aṭrāf method in his assemblies. The method was used by Aḥmad b. Ḥanbal, who traveled to Madīnah to convey the reports recorded with the mentioned method of his teacher Abū Alḳama al-Farawī and at the same time listen to the related transfers from him, but who could not reach his goal. Therefore, the related narrations show that aṭrāf method was used effectively by ḥadīth scholars to remember the ḥadīths in the 1st and 2nd centuries, so sections from the narrations were written during the preparatory stage before coming to the scholarly assembly. Ultimately, the importance of these works, which manifest in different ways and offer great contributions to writing, is an undeniable reality for the history of ḥadīth.Aṭrāf, the plural form of taraf that literally means the side and edge of something, mean a section, phrase or first part that indicates the whole of the ḥadīth. This method, along with the formation of ḥadīth literature, has been started to be given as a name to the works organized alphabetically according to the name of the Companions or ḥadīth texts by mentioning some of the ḥadīths with the 4th century. In this sense, aṭrāf has started to become a unique literature in a sort of taṣnīf and takhrīj method in order to make use of the ḥadīth writing easier.Aṭrāf studies may aim to bring together the isnāḍs (chains of transmitters) in all books, or they can be limited to the chains in certain books. Aṭrāf books contain the ḥadīths in the Ṣaḥīḥān or Kutub al-Sittah and are written in alphabetical form based on the name of the companions or according to the texts of ḥadīth. The first of these, which has the attribute of the musnad in a sense, is arranged in alphabetical order according to the name of the Companion, and if there are many people who narrate from him, according to the names of the tābiīn and taba-i tābiīn (successors of the tābiīn). Accordingly, the narrations of each companion were determined and listed in al-phabetical order in aṭrāfs in musnad-type, and a passage was written from the ḥadīths and marked as books/chapters in the places of relevant narration. On the other hand, the second type of aṭrāf books were ar-ranged by giving a few words from a section that allows remembering the head or all of the ḥadīths without in-cluding chains, and then placing these texts in alphabetical order.The type of aṭrāf offers the opportunity to see different chains of narrations together and provides many facilities for the ḥadīth researchers. Therefore, ḥadīth researchers show great interest in this genre. Aṭrāfs provide many benefits for both ḥadīth chain and text. In this respect, it is possible to talk about many advantages of the ḥadīth to its transmister such as the determination of mutawātīr, garīb, azīz, mashhūr, mutābī or the testimony of ḥadīth; to provide facility to make a reputation research of a riwāyah (narration), to provide an opportunity to determine the location of the narrations in the primary sources; to detect the link or any blackout in the chain, to reach the nu-merical knowledge of ḥadīth narrated by a companion or other narrators, to obtain information about narrators whose identity is not explicitly stated, to know the first narrator, to provide the protection of the ḥadīths from taḥrīf or alteration, to give the opportunity to identify the subject they contain, to be able to obtain text integrity by getting various chains, the detection of the sabab al-wurūd (the context of ḥadīth).Due to the contributions we have mentioned above, many books on aṭrāf literature that started with Abu’l-Ḥasen Ali b. Omar ad-Dārakutnī have been published. Among them Aṭrāfu’s-ṣaḥīḥayn/el-Jem`u bayna’s-ṣaḥīḥayn by Abū Mas`ūd Ibrāhīm b. Muḥammad b. Ubayd ad-Dimashkī, Aṭrāfu’s-ṣaḥīḥayn by Abū Muḥammad Ḥalaf b. Muḥammad b. Ali al-Vasiṭī, Zaḫāiru’l-mavārīs fi’d-dalālati alā mavāzi`i’l-hadīs by Abdülganī b. İsmāīl en-Nāblūsī and Tuḥfatu’l-ashrāf bi ma`rifati’l-aṭrāf by Abu’l-Ḥajjāj Yūsuf b. Abdurrahman el-Mizzī can be mentioned. Some of the related works are arranged in musnad style while others are arranged according to the first word of ḥadīth texts. As a result, the method of aṭrāf has become a kind of indexer in order to facilitate reaching the source of the re-ports and make the highest level of use required by the classification and bookings of the ḥadīths in the 3rd centu-ry. These works, arranged alphabetically in the name of the Companions or according to the texts of the ḥadīth, have taken their place in the history of the ḥadīth with their many examples, both in the form of taṣnīf and the method of takhrīj by mentioning a part of the ḥadīth by the head. (shrink)