Let/. It :0 ---0 G be any two self maps of a compact connected oriented Lie group G. In this paper, for each positive integer k , we associate an integer with fk,hi . We relate this number with Lefschetz coincidence number. We deduce that for any two differentiable maps f, there exists a positive integer k such that k 5.2+1 , and there is a point x C G such that ft (x) = (x) , where A is the rank of G . Introduction Let G be an n-dimensional com -pact connected Lie group with multip-lication p ( .e 44:0 xG--+G such that p ( x , y) = x.y ) and unit e . Let [G, G] be the set of homotopy classes of maps G G . Given two maps f , f G ---• Jollowing [3], we write f. f 'to denote the map G-.Gdefined by 01.11® =A/WO= fiat® ,sea Given a point g EC and a differ-entiable map F: G G , write GA to denote the tangent space of G at g [4,p.10] , and denote by d x F the linear map rig F :Tx0 T, (x)G induced by F , it is called the differential of Fat g [4,p.22]. Let LA, Rx :0 G be respec-tively the left translation Lx(i)=4..(g,e) , and the right translation Rx(1)./..(gcg). Then there is a natural homomorphism Ad ,the adjoin representation, from G to GL(G•), (the group of nonsingular linear transformations of Qdefined as follows:- Ad(g)= deRe, od,Lx. Note that d xRc, ad.; =d(4,( Lx(e)))0 de; =d.(4, 04)=4(40 Re) = d(4(4, (e)))0 (44, =d ar, o (44, . Since G is connected , the image of Ad belongs to the connected component of G(G)containing the identity,i.e. for each g E 0, detAd(g) > 0 . By Exercise Al • Dr.-Prof.-Department of Mathematics- College of Science- University of Baghdad. •• Dr.-Department of Mathematics- College of Science for Woman- University of Baghdad.
the most important purposes and uses of the test results in the educational sector. This is because the quality of tests is related to their ability to predict the learner's behavior in the future, and the accuracy of the educational and administrative decisions that are taken in light of their results. The study aimed accordingly to reveal the predictive ability of the university Grade Point Average (GPA) in the Score of the specialized test for the position of teacher in the Ministry of Education in the Sultanate of Oman. It further aimed to investigate the differences in the predictive ability according to the specialization and academic year using the descriptive approach. The sample of the study consisted of (349) s/he students enro
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he dairy industry is one of the industrial activities classified within the food industries in all phases of the dairy industry, which leads to an increase in the amount of wastewater discharged from this industry. The study was conducted in the Abu Ghraib dairy factory, classified as one of the central factories in Iraq, located in the west of Baghdad governorate, with a design capacity of 22,815 tons of dairy products. The characteristics of the liquid waste generated from the factory were determined for the following parameters biological oxygen demand (BOD5), Chemical oxygen demand (COD), total suspended solids (TSS), pH, nitrate, phosphate, chloride, and sulfate with an average value of (1079, 1945, 323, 9.2, 24, 2
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Background: We aimed to investigate the accuracy of salivary matrix metalloproteinases (MMP)-8 and -9, and tissue inhibitor of metalloproteinase (TIMP)-1 in diagnosing periodontitis and in distinguishing periodontitis stages (S)1 to S3. Methods: This study was a case–control study that included patients with periodontitis S1 to S3 and subjects with healthy periodontia (controls). Saliva was collected, and then, clinical parameters were recorded, including plaque index, bleeding on probing, probing pocket depth, and clinical attachment level. Diagnosis was confirmed by assessing the alveolar bone level using radiography. Salivary biomarkers were assayed using an enzyme-linked immunosorbent assay. Results: A total of 45 patients (15
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The OpenStreetMap (OSM) project aims to establish a free geospatial database for the entire world which is editable by international volunteers. The OSM database contains a wide range of different types of geographical data and characteristics, including highways, buildings, and land use regions. The varying scientific backgrounds of the volunteers can affect the quality of the spatial data that is produced and shared on the internet as an OSM dataset. This study aims to compare the completeness and attribute accuracy of the OSM road networks with the data supplied by a digitizing process for areas in the Baghdad and Thi-Qar governorates. The analyses are primarily based on calculating the portion of the commission (extr
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The OpenStreetMap (OSM) project aims to establish a free geospatial database for the entire world which is editable by international volunteers. The OSM database contains a wide range of different types of geographical data and characteristics, including highways, buildings, and land use regions. The varying scientific backgrounds of the volunteers can affect the quality of the spatial data that is produced and shared on the internet as an OSM dataset. This study aims to compare the completeness and attribute accuracy of the OSM road networks with the data supplied by a digitizing process for areas in the Baghdad and Thi-Qar governorates. The analyses are primarily based on calculating the portion of the commission (extra road) and
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The current study aimed to reveal the moral values available in the Arabic language textbook (Al-Mu’nes) for grades eleven and twelve for post-basic education in the Sultanate of Oman. The researcher adopted the descriptive approach using the sentence as measuring unit in the analysis process. The five fields of moral values and their subcategories values represented the categories of the analysis. Iterations of values and the percentages of iteration were calculated in those textbooks. Moreover and the analysis and stability of the process were confirmed.
After referring to previous studies, it was possible to de
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The Ge0.4Te0.6 alloy has been prepared. Thin films of Ge0.4Te0.6 has been prepared via a thermal evaporation method with 4000A thickness, and rate of deposition (4.2) A/sec at pressure 2x10-6 Torr. The A.C electrical conductivity of a-Ge0.4Te0.6 thin films has been studied as a function of frequency for annealing temperature within the range (423-623) K, the deduced exponent s values, was found to decrease with increasing of annealing temperature through the frequency of the range (102-106) Hz. It was found that, the correlated barrier hopping (CBH) is the dominant conduction mechanism. Values of dielectric constant ε1 and dielectric loss ε2 were found to decrease with frequency and increase with temperature. The activation energies have
... Show MoreTopological indices provide important insights into the structural characteristics of molecular graphs. The present investigation proposes and explores a creative graph on a finite group G, which is known as the RIG. This graph is designated as ΓRS G2(4) indicating a simple undirected graph containing elements of G. Two distinct ertices are regarded as nearly the same if and only if their sum yields a non-trivial involution element in G. RIGs have been discovered in various finite groups. We examine several facets of the RIG by altering the graph through the conjugacy classes of G. Furthermore, we investigate the topological indices as applications in graph theory applying the distance matrix of the G2(4) group.