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.
Fire incidences are classed as catastrophic events, which mean that persons may experience mental distress and trauma. The development of a robotic vehicle specifically designed for fire extinguishing purposes has significant implications, as it not only addresses the issue of fire but also aims to safeguard human lives and minimize the extent of damage caused by indoor fire occurrences. The primary goal of the AFRC is to undergo a metamorphosis, allowing it to operate autonomously as a specialized support vehicle designed exclusively for the task of identifying and extinguishing fires. Researchers have undertaken the tasks of constructing an autonomous vehicle with robotic capabilities, devising a universal algorithm to be employed
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The Umm Al-Naaj Marsh was chosen in Maysan province, and it is one of the sections of Mar Al-Hawza, which is one of the most prominent Iraqi marshes in the south. The marshes are located between latitudes 30 35 and 32 45 latitudes and longitudes 13 46 and 48 00. The area of the study area is 76479.432142 hectares to evaluate soil quality and health index and their spatial distribution based on measuring physical, chemical, biological and fertility traits and calculating the total quality index for those characteristics. Using an auger drilling machine, we collected 50 randomly selected surface samples, evenly distributed across the study region, from Al-Aq 0.0–0.30 m, noting their precise locations along the way. Soil health and quality w
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The research discussed the possibility of adsorption of Brilliant Blue Dye (BBD) from wastewater using 13X zeolite adsorbent, which is considered a byproduct of the production process of potassium carbonate from Iraqi potash raw materials. The 13X zeolite adsorbent was prepared and characterized by X-ray diffraction that showed a clear match with the standard 13X zeolite. The crystallinity rate was 82.15% and the crystal zeolite size was 5.29 nm. The surface area and pore volume of the obtained 13X zeolite were estimated. The prepared 13X zeolite showed the ability to remove BBD contaminant from wastewater at concentrations 5 to 50 ppm and the removal reached 96.60% at the lower pollutant concentration. Adsorption measurements versus tim
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For criminal investigations, fingerprints remain the most reliable form of personal identification despite developments in other fields like DNA profiling. The objective of this work is to compare the performance of both commercial charcoal and activated carbon powder derived from the Alhagi plant to reveal latent fingerprints from different non-porous surfaces (cardboard, plain glass, aluminum foil sheet, China Dish, Plastic, and Switch). The effect of three variables on activated carbon production was investigated. These variables were the impregnation ratio (the weight ratio of KOH: dried raw material), the activation temperature, and the activation time. The effect factors were investigated using Central Composite Design
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The investigation of determining solutions for the Diophantine equation over the Gaussian integer ring for the specific case of is discussed. The discussion includes various preliminary results later used to build the resolvent theory of the Diophantine equation studied. Our findings show the existence of infinitely many solutions. Since the analytical method used here is based on simple algebraic properties, it can be easily generalized to study the behavior and the conditions for the existence of solutions to other Diophantine equations, allowing a deeper understanding, even when no general solution is known.
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In this work the interaction of plasma jet with water and hydrogen peroxide liquids used for assisted teeth bleaching by plasma jet had been study. A homemade plasma jet system was used. The plasma jet supply by 15 W electrical power generated by high voltage power supply of 9.6 kV peak to peak and frequency of 33 kHz .this power supply generate high electric field on electrodes that would be enough to ionize the argon gas. Some important agents were study such as the effect of the Ar gas flow rates on the length of the plasma jet, the influence of plasma jet on some properties of water and two hydrogen peroxide concentrations 25 % and 30 % like pH, conductivity and liquid temperature for different exposure time. The liquids temperature
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