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.
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Importance of study :
The interesting of industrial sector and developing it after corner stone. It is important to build economic balanced and get behind and front linkage for transformative industrials, and this is important to get balanced.
The value of transformative industry sector is more important than other sectors in economy, and this is reflect the capital in industrial sector .Industry sector is larger than other sectors ,so it regards the motive which moving the economic wheel .
The problem :
The research treats reduction and retreat the ind
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Innovative laboratory research and fluid breakthroughs have improved carbonate matrix stimulation technology in the recent decade. Since oil and gas wells are stimulated often to increase output and maximum recovery, this has resulted in matrix acidizing is a less costly alternative to hydraulic fracturing; therefore, it is widely employed because of its low cost and the fact that it may restore damaged wells to their previous productivity and give extra production capacity. Limestone acidizing in the Mishrif reservoir has never been investigated; hence research revealed fresh insights into this process. Many reports have stated that the Ahdeb oil field's Mishrif reservoir has been unable to be stimulated due to high injection pressures, wh
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Sustainability is a major demand and need pursued by cities in all areas of life due to the environmental, social and economic gains they provide, especially in the field of city planning and urban renewal projects that aim to integrate the past, present and future.
The research aims to evaluate the Haifa Street renewal project, and Al-Shawaka district, one of the Baghdad districts located next to Al-Karkh, was elected by comparing the sustainability indicators of urban renewal with the reality of the situation through a field survey and questionnaire form and focusing on the social and economic impacts and environmental for the project on the study area. To reach the most important conclusions and recommendations
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An optoelectronic flow-through detector for active ingredients determination in pharmaceutical formulations is explained. Two consecutive compact photodetector’s devices operating according to light-emitting diodes-solar cells concept where the LEDs acting as a light source and solar cells for measuring the attenuated light of the incident light at 180˚ have been developed. The turbidimetric detector, fabricated of ten light-emitting diodes and five solar cells only, integrated with a glass flow cell has been easily adapted in flow injection analysis manifold system. For active ingredients determination, the developed detector was successfully utilized for the development and validation of an analytical method for warfarin determination
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Innovative laboratory research and fluid breakthroughs have improved carbonate matrix stimulation technology in the recent decade. Since oil and gas wells are stimulated often to increase output and maximum recovery, this has resulted in matrix acidizing is a less costly alternative to hydraulic fracturing; therefore, it is widely employed because of its low cost and the fact that it may restore damaged wells to their previous productivity and give extra production capacity. Limestone acidizing in the Mishrif reservoir has never been investigated; hence research revealed fresh insights into this process. Many reports have stated that the Ahdeb oil field's Mishrif reservoir has been unable to be stimulated due to high inj
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Hygienic engineering has dedicated a lot of time and energy to studying water filtration because of how important it is to human health. Thorough familiarity with the filtration process is essential for the design engineer to keep up with and profit from advances in filtering technology and equipment as the properties of raw water continue to change. Because it removes sediment, chemicals, odors, and microbes, filtration is an integral part of the water purification process. The most popular technique for treating surface water for municipal water supply is considered fast sand filtration, which can be achieved using either gravity or pressure sand filters. Predicting the performance of units in water treatment plants is
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Nonlinear differential equation stability is a very important feature of applied mathematics, as it has a wide variety of applications in both practical and physical life problems. The major object of the manuscript is to discuss and apply several techniques using modify the Krasovskii's method and the modify variable gradient method which are used to check the stability for some kinds of linear or nonlinear differential equations. Lyapunov function is constructed using the variable gradient method and Krasovskii’s method to estimate the stability of nonlinear systems. If the function of Lyapunov is positive, it implies that the nonlinear system is asymptotically stable. For the nonlinear systems, stability is still difficult even though
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