This study presents a mathematical model describing the interaction of gut bacteria in the participation of probiotics and antibiotics, assuming that some good bacteria become harmful through mutations due to antibiotic exposure. The qualitative analysis exposes twelve equilibrium points, such as a good-bacteria equilibrium, a bad-bacteria equilibrium, and a coexisting endemic equilibrium in which both bacteria exist while being exposed to antibiotics. The theory of the Sotomayor theorem is applied to study the local bifurcation around all possible equilibrium points. It’s noticed that the transcritical and saddle-node bifurcation could occur near some of the system’s equilibrium points, while pitchfork bifurcation cannot be accrued at

The importance of the research lies in knowing the effect of the exercises of the reciprocal method in developing some physical abilities in learning the performance of the players for the effectiveness of the long jump in an economical manner in terms of time and effort and knowing their positive impact and the extent of their impact in creating the required learning for students, and the research aims to prepare reciprocal style exercises in developing some abilities The researchers used the experimental method in the pre and post test for the experimental and control groups to suit the nature of the research, and the research community was identified for the long jump players, the Specialized School for Talent Care in the 2022 sports sea

 Several studies have indicated an unprecedented increase in the number of Arab youth who watch music videos. It is also a custom in Arab countries to broadcast songs at their happy parties such as weddings, engagements and birthdays. We see that guests and party owners interact by dancing and singing with the songs, while the viewership rates of Arab music videos have reached millions on YouTube. The researcher decided to study the image of women through the lyrics of these songs, due to their importance in shaping the image of women in the minds of young people and shaping the (self-image) of young women. Twenty songs were selected from the most watched songs on YouTube for the year 2024, and it was found that the negative qualities of t

Knowing the nature of the relationship in federal systems lies in studying the political, social, economic and cultural forces, for their role in laying the foundations and the federal system. No matter how important the pillars of that system are, and through this research, we will learn how the political elites were able to crystallize this unique system, until it became one of the political systems that some countries seek to copy and apply its experience.

The first section of this research discussed  the manner of the research from many sides like the problem it faces, importance of it , its targets ,boundaries, the way to collect and get information's and its assumption.

When the second chapter discussed the press – manufacturing and the development ,importance and types of newspapers, also its merits and weaknesses.

The third chapter talked about the scientific side and how to choose an assumption for the research . as it talked also about the apparent honest and stability tests that help in analyzing the research until getting results and so the right assumption for  the research will be chosen.

And finally, the fourth chapter put highlight on the be

ABSTRACT

Background : The aim of this work is to assess the role of breast sonography  and ductography in the evaluation of different causes of nipple discharge.

Methods :  The study will be carried out on twenty-five female patients referred to the Radiodiagnosis department at Alexandria Main University Hospital presenting with nipple discharge.

They were divided into two groups:

Group I include 10 patients (40%) with surgically significant nipple discharge who were the patients with unilateral, uniorificial surgically significant colour type nipple discharge .They were investigated by mammography, sonography, and ductography.

Group II include 15 patients

Improvisation is that spontaneous automatic achievement which is formed by a new cognition that is not based on something prior to it or a previous cognition. It is instantaneous. Whereas in art, improvisation also inters in all types of applied and performance arts as a foundation for launching and initiating it in music, painting, cinema, television and theater. In order to study the improvisation of the actor, the researcher put forward a theoretical study that included two sections. The first section is (the improvisation concept) and the second section is (improvisation in the show). The researcher, in the research procedures, took an intentional sample that was represented by the theatrical show (Rehearsal in Hell Play) and after t

The launch of the EU’s Eastern Partnership in 2009 intended to signal a new, elevated level of EU engagement with its Eastern neighborhood. Yet there remain several long-simmering and potentially destabilizing conflicts in the region, with which EU engagement thus far has been sporadic at best. The Union’s use of its Common Security and Defense Policy (CSDP) in the region and to help solve these disputes has been particularly ad hoc and inconsistent, wracked by inter-institutional incoherence and undermined by Member States’ inability to agree on a broad strategic vision for engagement with the area.

The three CSDP missions deployed to the region thus far have all suffered from this incoherence to various extents. In particu

There are many reasons made Basra a good defense, first of all was the
establishment of the famous Arab tribes, like Bakir bn. Wail and Tamim. Besides there were
A lot the armed forces. In addition to that the strategic place of Basra. All these reasons
played a good part when the war broke out.
The Arabic tribes played a good part in the war to engaged Persians, and prevent
them from sending any helps to their armies against the Kufain who were fighting Persians in
the North of Iraq. While the Basrain army aiming to Conquer the south of Iraq.
Basra become the first strategic place for the Islamic movement of the conquers.
Latter on Kufa and Bahrain were the second and third.
The Muslims guranted the pupils of