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The Product of Automorphic Weighted Composition Operators on Hardy Space H <sup>2</sup>
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Abstract<p>Let <inline-formula> <tex-math><?CDATA $n\in {\mathbb{N}},{p}_{i}\in {\rm{U}},{\alpha }_{{P}_{i}}(z)=\frac{{p}_{i}-z}{1-{\bar{p}}_{i}z}(z\in {\rm{U}})$?></tex-math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mrow> <mi>n</mi> <mo>∈</mo> <mi>ℕ</mi> <mo>,</mo> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>∈</mo> <mi mathvariant="normal">U</mi> <mo>,</mo> <msub> <mi>α</mi> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>−</mo> <mi>z</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mover accent="true"> <mi>p</mi> <mo>¯</mo> </mover> <mi>i</mi> </msub> <mi>z</mi> </mrow> </mfrac> <mo stretchy="false">(</mo> <mi>z</mi> <mo>∈</mo> <mi mathvariant="normal">U</mi> <mo stretchy="false">)</mo> </mrow> </math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JPCS_1530_1_012045_ieqn1.gif" xlink:type="simple"></inline-graphic> </inline-formula>, and let <inline-formula> <tex-math><?CDATA ${f}_{1}\in {H}^{\infty },i=1,\ldots,n$?></tex-math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>∈</mo> <msup> <mi>H</mi> <mi>∞</mi> </msup> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mrow> </math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JPCS_1530_1_012045_ieqn2.gif" xlink:type="simple"></inline-graphic> </inline-formula>. We discuss the relation between the points <italic>p<sub>i</sub> </italic> in U and the functions <italic>f<sub>i</sub> </italic> in U and the properties of the product of automorphic weighted composition operators <inline-formula> <tex-math><?CDATA ${W}_{{f}_{1},{\alpha }_{{p}_{1}}}\,{W}_{{f}_{2},{\alpha }_{{p}_{2}}}\ldots {W}_{{f}_{i},{\alpha }_{pi}}$?></tex-math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mrow> <msub> <mi>W</mi> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </msub> <mspace width="0.25em"></mspace> <msub> <mi>W</mi> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </msub> <mo>…</mo> <msub> <mi>W</mi> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow> <mi>p</mi> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mrow> </math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JPCS_1530_1_012045_ieqn3.gif" xlink:type="simple"></inline-graphic> </inline-formula> on Hardy space H<sup>2</sup>. In fact, it is very nice connection between analytic function theory and operator theory. In this paper, we give the sufficient and necessary conditions to be normal, unitary, hermitian operator on <italic>H</italic> <sup>2</sup> and we shall present the shape of the numerical range of it.</p>
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Publication Date
Wed Jun 30 2010
Journal Name
Al-kindy College Medical Journal
Frozen Shoulder in Type 2 Diabetes Mellitus
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Background: Frozen shoulder affects 2-5% of the
general population, and around 10-30% of diabetic
patients. It affect mainly the non-dominant shoulder,
and has more incidence in patients with poor
glycemic control.
Objective: To detect the incidence of frozen
shoulder in type 2 diabetic patients attending the
Specialized Center for Endocrinology and Diabetes
in Baghdad.Patients and methods: One hundred
patients with frozen shoulder were included in the
study from a total number of 580 type 2 diabetics
over a period of six months. 70 patients were
females and 30 patient were males. All were
investigated for fasting blood
glucose and HbA1c.
Results: The non-dominant shoulder was
involved in

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Publication Date
Thu Aug 16 2012
Journal Name
Journal Of Chemical And Pharmaceutical Research
Synthesis, Characterization, Schiff Base Phenyl 2-(2-hydroxybenzylidenamino)benzoate and its complexes with LaIII,Cr III and Pr III
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The aim of the work is the synthesis and characterization of the tridentate Schiff base (HL) containing (N and O) as donor atoms type (ONO). The ligand is: (HL) phenyl 2-(2-hydroxybenzylidenamino)benzoate . This ligand was prepared by the reaction of (phenyl 2-aminobenzoate) with salicylaldehyde under reflux in ethanol and few drops of glacial acetic acid which gave the ligand (HL). The prepared ligand was characterized by (FT IR,UV–Vis) spectroscopy, Elemental analysis of carbon, hydrogen and nitrogen (C.H.N.) and melting point. The ligand was reacted with some metal ions under reflux in ethanol with (1 metal :2 ligand )mole ratio which gave complexes of the general formula: [M(L)2]Cl , M = Cr III La III and , Pr III Products were found

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Publication Date
Sun Nov 01 2020
Journal Name
Iop Conference Series: Materials Science And Engineering
SnO<sub>2</sub> doped In<sub>2</sub>O<sub>3</sub> thin films as reducing gas sensors
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Summery<p>This research includes description of the x-ray diffraction, morphology and sensing measurements of SnO<sub>2</sub> doped In<sub>2</sub>O<sub>3</sub> thin films synthesized by pulsed laser deposition method on glass and silicon wafer substrates. In2O3:SnO2 powders were obtained by mixing In2O3 with SnO2 in the desired ratio, and calcination the at temperature 1273 K for 5 hours. SnO<sub>2</sub> doped In<sub>2</sub>O<sub>3</sub> thin films with different ratios (0, 0.01, 0.03, 0.05, 0.07, and 0.09% wt.) were prepared using pulsed laser deposition method. The structural investigation using X-ray diffraction revealed that the mai</p> ... Show More
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Publication Date
Sat Jul 01 2017
Journal Name
Energy Procedia
HgBa 2 Ca n-1 Cu n O 2n+2+δ Superconducting thin films Prepared by Pulsed Laser Deposition
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Publication Date
Sun Sep 01 2013
Journal Name
Baghdad Science Journal
Synthesis, Spectral Studies, and Theoretical Treatment of New Ni(II),and Co(II) Complexes of Bidenetate Ligands 2-Benzamido Benzothiazole ,and 2-Actamido Benzothiazole
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New metal complexes of the ligands 2-benzamido benzothiazole(B1), and 2-actamido benzothiazole(B2) with metal ions Ni(II),and Co(II) were prepared in alcoholic medium. The prepared complexes were characterized by FT-IR and electronic spectroscopy, Magnetic susceptibility, Flame Atomic Absorption technique as well as elemental analysis and conductivity measurement. From the spectral studies, an octahedral monomer structure proposed for Ni(II) complexes, and a tetrahedral monomer structure for Co(II)complexes.Semi-empirical methods (PM3,and ZINDO/1)were carried out to evaluate the heat formation( ?H?f)binding energy(?Eb) and dipole moment(µ)for all metal complexes. Also vibration frequencies, Electrostatic potential, HOMO and LUMO

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Publication Date
Sat Jan 23 2016
Journal Name
Journal University Of Kerbala
Synthesis and Characterization of Some Metal Complexes of [2-chloro-N-(1,5-dimethyl-3-oxo-2-phenyl-2,3-dihydro-1Hpyrazol-4-ylcarbamothioyl)acetamide] (L)
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A new ligand [ 2-chloro-N- (1,5-dimethyl-3-oxo-2-phenyl-2,3-dihydro -1H-pyrazol- 4- ylcarbamothioyl)acetamide](L) was synthesized by reacting the Chloro acetyl isothiocyanate with 4-aminoantipyrine,The ligand was characterized by(C HNS) elemental microanalysis and the spectral measurements including Uv-Vis ,IR ,1H and13C NMR spectra, some transition metals complex of this ligand were prepared and characterized by Uv-Vis, FT-IR spectra, conductivity measurements, magnetic susceptibility and atomic absorption. From the obtained results the molecular formula of all prepared complexes were [M(L)2(H2O)2]Cl2 (M+2 =Mn, Co, Ni, Cu, Zn, Cd and Hg),the proposed geometrical structure for all complexes were octahedra

Publication Date
Thu Jul 24 2025
Journal Name
Desalination And Water Treatment
Removal of lead, cadmium, and copper from wastewater using Cinnamon bark waste to introduce it as a value-added product: Removal, kinetics and thermodynamics study
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Publication Date
Sat Oct 01 2022
Journal Name
Baghdad Science Journal
Absorption and Fluorescence Properties of 3,6-bis(5-bromothiophen-2-yl)-2,5-bis(2-ethylhexyl)-2,5-dihydropyrrolo[3,4-c]pyrrole-1,4-dione. A Covalent Fluorescence in Solution and in the Solid State
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In the cuurent article, the photophysical properties of 3,6-bis(5-bromothiophen-2-yl)-2,5-bis(2-ethylhexyl)-2,5-dihydropyrrolo[3,4-c]pyrrole-1,4-dione were investigated. The visible absorption bands at 527, 558 and 362 nm in propylene carbonate and the compound was found to be fluorescent in solution and in the plastic film with emission wavelengths between 550- 750 nm. The Stokes Shift of P.C., acetonitrile, diethyl ether, Tetrahydrofuran THF, cyclohexane, dibutyl ether, and dichloromethane DCM  are 734, 836, 668, 601, 601, 719, and 804 cm-1 in respectively. The Stokes Shift Δ was less in THF and cyclohexane, than the solvents, which indicates that the energy loss is less between the excitation and fluorescence states. The

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Publication Date
Tue Jan 09 2018
Journal Name
Iraqi Journal Of Cancer And Medical Genetics
Photodynamic Action of low power He-Ne laser on photosensitized human Hep-2 and AMN3 cell lines with Hematoporphyrin derivative in vitro
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Photodynamic Action (PDA) by using appropriate wavelength of irradiation conjugated with porphyrin derivatives is a powerful mechanism of tumor destruction. Hematoporphyrin derivative has been shown to selectively localize in neoplastic cells and then cause destruction of them by generation of singlet oxygen when activated by low power He-Ne laser. Light which used in this study has been emitting from this laser has a wavelength equal to 632.8 nm (red light). Doses of laser had been varied from 3.6 J/cm2 to 14.4 J/cm2 . The beam of laser adjusted with a modified tissue culture plate. Cell lines had exposed to Hematoporphyrin D (HpD) for 24 hours before Laser exposure, their concentrations were varied from 5 µg/ml to 80 µg/ml. Resu

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Publication Date
Mon Jun 05 2023
Journal Name
Journal Of Engineering
Removal of Cu2+, Pb2+ , And Ni 2+ Ions From Simulated Waste Water By Ion Exchange Method On Zeolite And Purolite C105 Resin
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The removal of heavy metal ions from wastewater by ion exchange resins ( zeolite and purolite C105), was investigated. The adsorption process, which is pH dependent, shows maximum removal of metal ions at pH 6 and 7 for zeolite and purolite C105 for initial metal ion
concentrations of 50-250 mg/l, with resin dose of 0.25-3 g. The maximum ion exchange capacity was found to be 9.74, 9.23 and 9.71 mg/g for Cu2+, Pb2+, and Ni2+ on zeolite respectively, while on purolite C105 the maximum ion exchange capacity was found to be 9.64 ,8.73 and 9.39 for Cu2+, Pb2+, and Ni2+ respectively. The maximum removal was 97-98% for Cu2+ and Ni2+ and 92- 93% for Pb2+ on zeolite, while it was 93-94% for Cu2+, 96-97% for Ni2+, and 87-88% for Pb2+ on puroli

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