This paper contains an equivalent statements of a pre-  space, where  are considered subsets of with the product topology. An equivalence relation between the preclosed set  and a pre-  space, and a relation between a pre-  space and the preclosed set  with some conditions on a function  are found. In addition, we have proved that the graph  of  is preclosed in if  is a pre-  space, where the equivalence relation  on  is open.

     On the other hand, we introduce the definition of a pre-stable ( pre-stable) set by depending on the concept of a pre-neighborhood, where we get that every stable set is pre-stable. Moreover, we obtain that

     Various document types play an influential role in a lot of our lives activities today; hence preserving their integrity is an important matter. Such documents have various forms, including texts, videos, sounds, and images.  The latter types' authentication will be our concern here in this paper. Images can be handled spatially by doing the proper modification directly on their pixel values or spectrally through conducting some adjustments to some of the addressed coefficients. Due to spectral (frequency) domain flexibility in handling data, the domain coefficients are utilized for the watermark embedding purpose. The integer wavelet transform (IWT), which is a wavelet transform based on the lifting scheme,

A new algorithm is proposed to compress speech signals using wavelet transform and linear predictive coding. Signal compression based on the concept of selecting a small number of approximation coefficients after they are compressed by the wavelet decomposition (Haar and db4) at a suitable chosen level and ignored details coefficients, and then approximation coefficients are windowed by a rectangular window and fed to the linear predictor. Levinson Durbin algorithm is used to compute LP coefficients, reflection coefficients and predictor error. The compress files contain LP coefficients and previous sample. These files are very small in size compared to the size of the original signals. Compression ratio is calculated from the size of th

The Gender study is consider one of the concepts which the Postmodernism reached
after the end of Modernism, where the first one has limited the criticism study choices before
the second after closed many doors of subjects which was enriched by researches.
It is pretty clear that the root of this concept belongs to the Linguistics which provided
the Criticism with a countable reasons of it is growth.
The attention in the study of gender in Feminine Literature and Criticism increased in
Arabic studies since the early years of twenty one century, so this research is presented to be
an introduction to this subject which could pave the way to more studies.
In addition to the Gender studies this research deals with ano

Our aim in this paper is to introduce the notation of nearly primary-2-absorbing submodule as generalization of 2-absorbing submodule where a proper submodule  of an -module  is called nearly primary-2-absorbing submodule if whenever , for , , ,  implies that either  or  or . We got many basic, properties, examples and characterizations of this concept. Furthermore, characterizations of nearly primary-2-absorbing submodules in some classes of modules were inserted. Moreover, the behavior of nearly primary-2-absorbing submodule under -epimorphism was studied.

Background: Pyogenic discitis most frequently occurs
after home delivery, hemorridectomy, and dilatation and
curettage. Discitis is generally due to blood borne bacterial
invasion of the disc from adjacent end-plate via
communicating vessels. Infective discitis remains an
uncommon, but potentially serious cause for back pain.
Delayed diagnosis can occur and a high index of suspicion
may occur.
Methods: The study included 30 patients suffering from
localized low back pain with limitation of movement.
Diagnosis was made by history, physical examination and
investigation mainly E.S.R and MRI.
Results: 50.3% of patient with discitis had a history of
vaginal delivery, 15% had a history of hemorridectomy

Starting from bis (4,4'-diamino phenoxy) ethan(1), a variety of phenolicschiff bases (methylolic, etheric, epoxy) derivatives have been synthesized. All proposed structure were supported by FTIR, 1H-NMR, 13C-NMR Elemental analysis, some derivatives evaluated by thermal analysis (TGA).

 The problem of the research came through the following question (what is the effectiveness of active design in contemporary industrial products), and it emphasized the importance of research to make society with its members in a positive state of activity, and what can help the industrial product design to achieve this, but the goal of the study is (determining the characteristics and principles of design Active and its effectiveness in the industrial product), and the study was determined by industrial products related to daily, recreational and sports activity, The researcher relied on a form to determine the axes of analysis for the sample models, in order to reach the results of the research, the most important of which are: Realiz