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.
In this research, the preparation of bidentate Schiff base was carried out via the condensation reaction of both the salicylaldehyde with 1-phenyl-2,3-dimethyl-4-amino-5-oxo-pyrazole to form the ligand (L). The mentioned ligand was used to prepare complexes with transition metal ions Mn(II), Co(II), Ni(II), Cu(II) and Zn(II). The resulting complexes were separated and characterized by FTIR and UV-Vis spectroscopic technique. Elemental analysis for Carbon, Hydrogen and Nitrogen elements, electronic spectra of the ligand and complexes were obtained, and the magnetic susceptibility tests were also achieved to measure the dipole moments. The molar conductivities were also measured and determination of chlorine content in the complexes and
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The ligand 4-(2-aminmo-5-nitro-phenylazo)-1,5-dimethyl-2-phenyl-1,2-dihydro-pyrazol-3-one derived from 4-aminoantipyrine and 4-nitroaniline was synthesized. The synthesized ligand was characterized by 1HNMR, FT-IR, UV-Vis spectra and (C.H.N) analysis. Complexes of (YIII and LaIII ) with the ligand were prepared in aqueous ethanol with a 1:2 M:L ratio and at optimum pH. The prepared complexes were characterized by using flame atomic absorption, FT-IR, UV-Vis spectra,(C.H.N) analysis and conductivity measurement. The stoichiometry of complexes was studied by the mole ratio and job methods. A concentration range (1×10-4 - 3×10-4 M) obeyed Beer's law, the complex solutions show high values of molar absorption. On the basis of physicochemical
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Coupling reaction of 2-amino benzoic acid with the 8-hydroxy quinoline gave the azo ligand (H2L): 5-(2-benzoic acid azo )-8-hydroxy quinoline.Treatment of this ligand with some metal ions (CoII, NiII and CuII ) in ethanolic medium with a (1:2) (M:L) ratio yielded a series of neutral complexes with general Formula[M(HL)2],where: M=Co(II), Ni(II) and Cu(II), HL=anion azo ligand (-1).The prepared complexes were characterized using flame atomic absorption,FT-IR and UV-Vis spectroscopic methods as well as magnetic susceptibility and conductivity measurements.
. New Schiff base ligand 2-((4-amino-5-(3, 4, 5-trimethoxybenzyl) pyrimidin2-ylimino) (phenyl)methyl)benzoic acid] = [HL] was synthesized using microwave irradiation trimethoprim and 2-benzoyl benzoic acid. Mixed ligand complexes of Mn((ІІ), Co(ІІ), Ni(ІІ), Cu(ІІ), Zn(ІІ) and Cd(ІІ) are reacted in ethanol with Schiff base ligand [HL] and 8-hydroxyquinoline [HQ] then reacted with metal salts in ethanol as a solvent in (1:1:1) ratio. The ligand [HL] is characterized by FTIR, UV-Vis, melting point, elemental microanalysis (C.H.N), 1H-NMR, 13C-NMR, and mass spectra. The mixed ligand complexes are characterized by infrared spectra, electronic spectra, (C.H.N), melting point, atomic absorption, molar conductance and magnetic m
... Show MoreThe formation of Co(II), Ni(II), Cu(II), Zn(II), and Cd(II)-complexes (C1-C5) respectively was studied with new Schiff base ligand [benzyl(2-hydroxy-1-naphthalidene) hydrazine carbodithioate derived from reaction of 2-hydroxy-1-naphthaldehyde and benzyl hydrazine carbodithioate. The suggested structures of the ligand and its complexes have been determined by using C.H.N.S analyzer, thermal analysis, FT-IR, U.V-Visible, 1HNMR, 13CNMR , conductivity measurement , magnetic susceptibility and atomic absorption. According to these studies, the ligand coordinates as a tridentate with metal ions through nitrogen atom of azomethane , oxygen atom of hydroxyl, and sulfur atom of thione
... Show MoreThe Co (II), Ni (II) ,Cu(II), Zn(II) ,Cd(II) and Hg(II) complexes of mixed of amino acid (L-Alanine ) and Trimethoprim antibiotic were synthesized. The complexes were characterized using melting point, conductivity measurement and determination the percentage of the metal in the complexes by flame (AAS). Magnetic susceptibility, Spectroscopic Method [FT-IR and UV-Vis]. The general formula have been given for the prepared mixed ligand complexes [M(Ala)2(TMP)(H2O)] where L- alanine (abbreviated as (Ala ) = (C5H9NO2) deprotonated primary ligand, L- Alanine ion .= (C5H8NO2-) Trimethoprim (abbreviated as (TMP ) = C10H11N3O3S M(II) = Co (II),Ni(II) ,Cu(II), Zn(II) ,Cd(II) and Hg(II). The results showed that the deprotonated L- Alanine b
... Show MoreThe ligand 4-(2-aminmo-5-nitro-phenylazo)-1,5-dimethyl-2-phenyl-1,2-dihydro-pyrazol-3-one derived from 4-aminoantipyrine and 4-nitroaniline was synthesized. The synthesized ligand was characterized by 1HNMR, FT-IR, UV-Vis spectra and (C.H.N) analysis. Complexes of (YIII and LaIII ) with the ligand were prepared in aqueous ethanol with a 1:2 M:L ratio and at optimum pH. The prepared complexes were characterized by using flame atomic absorption, FT-IR, UV-Vis spectra,(C.H.N) analysis and conductivity measurement. The stoichiometry of complexes was studied by the mole ratio and job methods. A concentration range (1×10-4 - 3×10-4 M) obeyed Beer's law, the complex solutions show high values of molar absorption. On the basis of physicochemical
... Show MoreBackground: In young adults, multiple sclerosis is a prevalent chronic inflammatory demyelinating condition. It is characterized by white matter affection, but many individuals also have significant gray matter involvement. A double-inversion recovery pulse (DIR) pattern was recently proposed to improve the visibility of multiple sclerosis lesions. Objective: To find out how well a DIR sequence, FLAIR, and T2-weighted pulse sequences can find MS lesions in the supratentorial and infratentorial regions. Methods: A total of 37 patients with established diagnoses of multiple sclerosis were included in this cross-sectional study. Brain MRI was done using double inversion recovery, T2, and FLAIR sequences. The number of lesions was count
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