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.
Fourier Transform-Infrared (FT-IR) spectroscopy was used to analyze gasoline engine oil (SAE 5W20) samples that were exposed to seven different oxidation times (0 h, 24 h, 48 h, 72 h, 96 h, 120 h, and 144 h) to determine the best wavenumbers and wavenumber ranges for the discrimination of the oxidation times. The thermal oxidation process generated oil samples with varying total base number (TBN) levels. Each wavenumber (400–3900 cm−1) and wavenumber ranges identified from the literature and this study were statistically analyzed to determine which wavenumbers and wavenumber ranges could discriminate among all oxidation times. Linear regression was used with the best wavenumbers and wavenumber ranges to predict oxidation time.
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In present project, new Schiff base of 4, 4'- (((1E, 1'E)-1,4-.phenylenebis- (methane-ylylidene))-bis-(azane-ylylidene)) bis-(5-(4-chlorophenyl) -4H -1,2,4-triazole-3-thione) (L3) has been synthesized by condensation of 4-amino-5-(4-chlorophenyl)-2,4-dihydro-3H-1,2,4-triazole-3-thione with benzene-1,4-dicarboxaldehyde. The new asymmetrical Schiff base (L3) used as a ligand to synthesize a new complex with Co(II), Ni(II), Cu(II), Pd(II), and Pt(IV) metal ions by 1:2 (Metal: ligand) ratio. New ligand and their complexes have been exanimated and Confirmed by Fourier-transform infrared (FT-IR), Ultraviolet-visible (UV-visible), Proton nuclear magnetic resonance (1HNMR), carbon13 nuclear magnetic resonance (13CNMR), carbon-hydrogen nitrogen sulf
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Art of legend was arisen out of pillars of theatre which casts light on those legends that were written to glorify powers and situations surpass level of human beings . So we find that legend was transformed through its doctrine sides over level of concept and realization on field of subjects where by which Greek theatre was founded . As soon as it was adopted by drama art as legendary ritual phenomena with its stories . With that modes of social , spiritual expressions were differentiated within communities , peoples over all different countries and civilizations . This explains us the apparent difference in cases of legends , alike if they were within thoughtful , social or cultural sides between civilizations . Since drama art is mirr
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One of the most important virulence factors in Pseudomonas aeruginosa is biofilm formation, as it works as a barrier for entering antibiotics into the bacterial cell. Different environmental and nutritional conditions were used to optimize biofilm formation using microtitre plate assay by P. aeruginosa. The low nutrient level of the medium represented by tryptic soy broth (TSB) was better in biofilm formation than the high nutrient level of the medium with Luria Broth (LB). The optimized condition for biofilm production at room temperature (25 °C) is better than at host temperature (37 °C). Moreover, the staining with 0.1% crystal violet and reading the biofilm with wavelength 360 are considered essential factors in
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Fine aggregates used for concrete works in Sulaymaniyah city frequently fail to meet the standard requirements for gradation and fineness modulus in cement concrete. This paper aims to critically evaluate gradation, fineness modulus, and clay contents of various natural sands produced and used for concrete work in the region. Sixteen field sand samples were collected from various sites in Darbandikhan (5 samples), Qalat Dizah (5 samples), Koysinjaq (5 samples), and Piramagroon (1 sample) confirming to ASTM D75. The field samples were parted into test specimens based on ASTM C702. Then, sieve analysis was carried out on the oven-dry test specimens in compliance with ASTM C136. The test results of fine aggregates wer
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In this research work, a simulator with time-domain visualizers and configurable parameters using a continuous time simulation approach with Matlab R2019a is presented for modeling and investigating the performance of optical fiber and free-space quantum channels as a part of a generic quantum key distribution system simulator. The modeled optical fiber quantum channel is characterized with a maximum allowable distance of 150 km with 0.2 dB/km at =1550nm. While, at =900nm and =830nm the attenuation values are 2 dB/km and 3 dB/km respectively. The modeled free space quantum channel is characterized at 0.1 dB/km at =860 nm with maximum allowable distance of 150 km also. The simulator was investigated in terms of the execution of the BB84 p
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Background: There are several diseases in the body following recovery from COVID-19 infection because this virus operates on human genes in various types of peripheral tissue in the human body. It penetrates host cells via Angiotensin-converting enzyme-2 receptors and may have effects on bone remodeling, leading to osteopenia or osteoporosis, which are characterized by low bone mineral density, resulting in diminished bone strength. Bone Alkaline Phpsphatase is an enzyme released into the bloodstream as a soluble homodimer after being cleaved by a phospholipase and can be utilized as a biomarker of bone development. Objective: This research was designed to investigate the alteration of bone homeostasis balance in Iraqi post-COVID-19
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