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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 Mar 10 2021
Journal Name
Baghdad Science Journal
Preparation of some azo compounds by diazotization and coupling of 2- amino -5 – thiol -1,3,4- thiadizaole
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2- amino -5- thiol-1,3,4- thiadiazole (S1) was prepared by cyclic locking of thiosemicarbazide in the presence of anhydrous sodium carbonate and CS2. diazotization of (S1) compound gave diazonium salt (S2) that reacts with different activated aromatic compounds to get the following azo compounds ,2 [(4- aminophenyl) diazenyl ] 1,3,4- thiazdiazole-5- thiol (S3) ,2-[4-amino- 1-naphthyl diazenyl] -1,3,4 – thiazdiazole-5-thiol (S4) , 3-amino-4-[(5- mercapto -1,3,4- thiadiazole -2-yl) diazenyl ] phenol(S5) ,1-[(5-mercapto-1,3,4-thiadiazole-2-yl) diazenyl] -2-naphthol (S6) , 5-{[4-(dimethylamino) phenyl] diazenyl}-1,3,4-thiadiazole-2- thiol(S7) ,5-{[4-(diethylamino) phenyl] diazenyl}-1,3,4- thiadiazole-2- thiol(S8) ,2- amino-5-[(5-mercapto-1,3

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Publication Date
Fri Sep 20 2024
Journal Name
Journal Of Nano Research
Impact of Copper Doping on Nanocrystalline SnO&lt;sub&gt;2&lt;/sub&gt; Thin Films Synthesized by Sol-Gel Coating and Chemical Bath Deposition for Gas Sensor Applications
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This work focuses on the preparation of pure nanocrystalline SnO2 and SnO2:Cu thin films on cleaned glass substrates utilizing a sol-gel spin coating and chemical bath deposition (CBD) procedures. The primary aim of this study is to investigate the possible use of these thin films in the context of gas sensor applications. The films underwent annealing in an air environment at a temperature of 500 C for duration of 60 minutes. The thickness of the film that was deposited may be estimated to be around 300 nm. The investigation included an examination of the structural, optical, electrical, and sensing characteristics, which were explored across various preparation circumstances, specifically focusing on varied

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Publication Date
Mon Oct 01 2018
Journal Name
Materials Research Bulletin
Fabrication of CdSe nanoparticles sensitized TiO 2 nanotube arrays via pulse electrodeposition for photoelectrochemical application
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Publication Date
Sun Dec 07 2014
Journal Name
Baghdad Science Journal
Estimation of Serum Osteocalcin Levels in Osteoporotic Postmenopausal Iraqi Women with Type 2 Diabetes Mellitus
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Background: Osteoporosis (OP) is a systemic disease characterized by low bone mass and micro architectural deterioration of bone tissue, resulting in an increased risk of fractures and has touched rampant proportions. Osteocalcin, one of the osteoblast-specific proteins, showed that its functions as a hormone improves glucose metabolism and reduces fat mass ratio. This study is aimed to estimate the osteocalcin and glucose level in blood serum of osteoporotic postmenopausal Women with and without Type 2 Diabetes.Materials and methods: 60 postmenopausal women with osteoporosis divided into two groups depending on with or without T2DM, 30 patients for each. Serum samples of 30 healthy postmenopausal women were collected as control group. Ost

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Publication Date
Thu Sep 26 2019
Journal Name
International Journal Of Pharmaceutical Research
Synthesis and identification of novel 2-thioxoimidazolidin- 4-one derivatives containing azo and ester groups
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The compounds 3-[4̄-(4˭-methoxybenzoyloxy) benzylideneamino]-2-thioxo-imidazolidine-4-one(3)aand 4-(1-(5-oxo- 2-thioxoimidazolidin-1-ylimino)ethyl)phenyl acetate(3)b were prepared from the reaction of aromatic aldehyde or ketone(1)a,bwith thiosemicarbazide to give aryl thiosemicarbazones(2)a,b ,followed by cyclization with ethylchloroacetate in the presence of fused sodium acetate. Treatment the compounds(3)a,bwith 4- hydroxybenzenediazoniumchloride yielded the correspondings4-((4-((4-hydroxyphenyl)diazenyl)-5-oxo-2- thioxoimidazolidin-1-ylimino)methyl)phenyl 4-methoxybenzoate(4)aand4-(1-(4-((4-hydroxyphenyl)diazenyl)-5-oxo-2- thioxoimidazolidin-1-ylimino)ethyl)phenyl acetate(4)b.The new 2-thioxo-imidazolidin-4-one with esters (5-7)a,b sy

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Publication Date
Mon Apr 01 2019
Journal Name
Biochemical And Cellular Archives
Prevalence of Hbv, Hcv and Hiv-1, 2 infections among patients in Saladin governorate, Iraq
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Hepatitis B and Hepatitis C viruses are the major health problem in the worldwide. In the Middle East, the prevalence of HBV in general population with the chronic infectionsis 2-5%,whereas the prevalence of HCV is about 1% in Arabian Gulf countries. World Health Organization (WHO) revealed that the risks of HBV and HCV transmissionas well as human immunodeficiency virus (HIV) through the transfusion of contaminated blood and blood products is high, because of the fragility of health services in these countries. Several viral diseases are transportby different modes like bloodtransfusion, sexual contact, and unsafe injections. The mostcommon blood-transmitted viruses are hepatitis B virus(HBV), hepatitis C virus (HCV) and humanimmunodeficie

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Publication Date
Sat Nov 04 2023
Journal Name
Iraqi Journal Of Pharmaceutical Sciences( P-issn 1683 - 3597 E-issn 2521 - 3512)
Synthesis, Characterization, Anti-Inflammatory, and Antimicrobial Evaluation of New 2-Pyrazolines Derivatives Derived from Guaiacol
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Abstract:         Chalcones were used to synthesis series of 2-pyrazoline derivatives and evaluated their antimicrobial and anti-inflammatory activities        (E)-1,3-diphenylprop-2-en-1-one (1-5) were synthesized by Claisen-Schmidt Condensation method through the reaction of acetophenone with five various para substituted benzaldehyde in presence of KOH, the reaction monitoring by TLC and the result intermediates were checked by melting point and FT-IR Various 2-Pyrazoline derivatives were prepared by one pot reaction that involved the refluxing of (E)-1,3-diphenylprop-2-en-1-one (1–5) and Hydrazine monohydrate in the presence of glacial acetic acid for 24 hours at a temperature of (45–50) °C fo

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Publication Date
Thu Aug 08 2024
Journal Name
New Materials, Compounds And Applications
MOLECULAR MODELLING AND THEORETICAL DESIGN OF NOVEL NIRMATRELVIR DERIVATIVES AS SARS-COV-2 ENTRY INHIBITORS
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The pandemic SARS-CoV-2 is highly transmittable with its proliferation among nations. This study aims to design and exploring the efficacy of novel nirmatrelvir derivatives as SARS entry inhibitors by adapting a molecular modeling approach combined with theoretical design. The study focuses on the preparation of these derivatives and understanding their effectiveness, with a special focus on their binding affinity to the S protein, which is pivotal for the virus’s access to the host cell. Considering molecular docking aspects in the scope of a study on nirmatrelvir derivatives and S protein, dynamics simulations with 25 nanoseconds of their binding are explored. The study shows that these derivatives might work as effective antivi

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Publication Date
Tue May 10 2011
Journal Name
Al-qadisiyah Journal For Science
Synthesis and Characterization of 2-amino -5-phenyl-1,3,4-Oxadiazole Complexes with Selected Metal Ions.
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The new bidentate ligand 2-amino-5-phenyl-1,3,4-oxadiazole (Apods) was prepared by the reaction of benzaldehyde semicarbazone with bromine and sodium acetate in acetic acid gave. The prepared ligand was identified by Microelemental Analysis, FT.IR, UV-Vis and 1HNMR spectroscopic techniqes. Treatment of the prepared ligand with the following selected metal ions (MnII, CoII, NiII, CuII and ZnII) in aqueous ethanol with a 1:2 M:L ratio, yielded a series of complexes of the general formula [M(L)2Cl2].The prepared complexes were characterized using flame atomic absorption, (C.H.N)Analysis, FT.IR and UV-Vis spectroscopic methods as well as magnetic susceptibility and conductivity measurements. Chloride ion content was also evaluated by Mohr metho

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Publication Date
Thu Jan 01 2015
Journal Name
Health
Evaluation of Myocardial Performance in Hypertensive Patients with Type 2 Diabetes and Normal Ejection Fraction
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