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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
Tue Jul 13 2010
Journal Name
Journal Of Kerbala University
Synthesis and Characterization of Fe(II),Cd(II),Hg(II) and Ag (I) complexes with 2-Thiotolylurea Ligand
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Complexes reaction of Fe+2, Cd+2, Hg+2 and Ag+ with the 2-thiotolylurea were prepared in ethanolic medium with the (1:1) M:L ratio yielded a series of neutral complexes. The prepared complexes were characterized using flame atomic absorption, micoelemental analysis (C.H.N), chloride content (Mohr Method) , FT.IR and UV-Vis spectroscopic, as well as magnetic susceptibility and conductivity measurement. From the above data, the proposed molecular structure for Fe+2, Cd+2 and Hg+2 complexes are tetrahedral geometry while Ag+ complex is trigonal structure.

Publication Date
Sun Sep 01 2013
Journal Name
Baghdad Science Journal
Synthesis and Characterization of [2-(carboxy methylene-amino)-phenyl imino] acetic acid (L) and its some metal complexes
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New Schiff base, namely [2-(carboxy methylene-amino)-phenyl imino] acetic acid (L) and its some metal complexes [LCo.2H2O], [LNi.2H2O], [LCu].3H2O, [LCd.2H2O], [LHg.2H2O] and [LPb.2H2O], were reported and characterized by elemental analysis, metal content, spectroscopic methods, magnetic moments and conductivity measurements, it is found that the geometrical structures of these complexes are octahedral [Co(II), Ni(II), Cd(II), Hg(II), Pb(II) and square planar Cu(II).The complexes have been found to posses 1:1 (M:L) stoichiometry

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Publication Date
Fri Apr 01 2022
Journal Name
Chemical Methodologies
Design, Synthesis of Imidazolone and Oxazepine Derivatives Bearing Imidazo (2, 1-b) Thiazole along with its Antimicrobial Activity
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A new methodology was applied to the synthesis of new imidazolones and oxyazepine derivatives containing imidazo thiazole fused rings. Starting with 5-(4-bromo phenyl) imidazo (2, 1-b) thiazole, which was synthesized using the standard procedure, the Carbaldehyed group was introduced at position 6 of 5-(4-bromo phenyl) imidazo (2, 1-b) thiazole. Then, this 6-carbaldehyed derivative was condensed with different substituted aromatic amines to afford new Schiff bases. The latter were cyclized into new oxazepine and imidazolone derivatives by using phthalic anhydride and glycine, respectively. These new derivatives were characterized by using FT-IR, 1HHNMR, and 13CNMR spectra, as well as examined (evaluated) for anti-bacterial and anti-fungal a

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Publication Date
Thu Oct 08 2015
Journal Name
Journal Of Kerbala University
Synthesis, Spectroscopic and Biological Studies of some metal ions complexes with 1-Phenyl-3-pyridin-2-ylmethyl-thiourea
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حضر الليكاند (L) 1-فنيل-3-بردين-2-يل مثيل-ثايويوريا من تفاعل 2-أمينو مثيل بردين مع فنيل ايزوثايوسيانيت وبنسبة 1: 1 وشخص الليكاند بواسطة التحليل الدقيق للعناصر (C, H, N), الأشعة تحت الحمراء، الأشعة فوق البنفسجية–المرئية وطيف الرنين النووي المغناطيسي كما حضرت وشخصت معقدات أملاح بعض ايونات العناصر الثنائية التكافؤ (Co, Ni, Cu, Cd and Hg). استخدمت تقنية الأشعة تحت الحمراء، الأشعه فوق البنفسجية-المرئية, التوصيلية الكهربائية و الا

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Publication Date
Thu Oct 20 2022
Journal Name
Ibn Al-haitham Journal For Pure And Applied Sciences
A study of Hepcidin levels and other Biochemical parameters in woman with Osteoporosis with Type 2 Diabetes Mellitus
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Background: Diabetes mellitus (DM) could be regarded as a set of chronic metabolic disorders which have a common aspect of hyperglycemia. The resistance in the peripheral actions of insulin or impaired insulin secretion could be the reason  hepcidin which is a peptide hormone derived from liver, in systemic iron homeostasis is an essential regulator, and its lopsided production participates in the pathogenesis of iron disorders in spectrum. Osteoporosis often accompanies many diseases like ß-thalassemia, hemochromatosis, sickle liver diseases, cell disease and hemosiderosis featured by iron overload, evidences suggest that Iron overload and iron deficiency are suggested by evidences that they affect bone in a negative way, acting

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Publication Date
Sat Feb 01 2020
Journal Name
Meta Gene
Interleukin-2 serum level, genetic polymorphism (rs2069763), anti-rubella antibody and risk of multiple sclerosis among Iraqi patients
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Background: Multiple sclerosis (MS) is a chronic neurodegenerative autoimmune disease mediated by autoreactive T cells against myelin-basic proteins. Cytokines are suggested to play a role in the etiopathogenesis of the disease. Among these cytokines is interleukin-2 (IL-2). Aim of the study: To investigate the association between IL2+166 G/T single nucleotide polymorphism (SNP: rs2069763) and MS in Iraqi patients. Serum level of IL-2 was also detected. Anti-rubella IgG antibody was further determined in the sera of patients. Patients and methods: Eighty MS patients (28 males and 52 females; age mean ± SD: 39.2 ± 16.1 years) and 80 healthy control matched patients for age (32.15 ± 16.13 years) and gender (28 males and 52 females) were en

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Publication Date
Sun Jan 01 2023
Journal Name
Journal Of Population Therapeutics And Clinical Pharmacology
IGG and IGM response in a group of Iraqi health care worker following SARS-CoV-2 mRNA vaccine
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Publication Date
Sun Jun 05 2016
Journal Name
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Synthesis and Characterization of New Heterocyclic Compounds from 2, 5- dimercapto -1, 3, 4-Thiadiazole and Their Resins
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In this research, a new 1, 3, 4-Thiadiazole derivatives have been synthesized by many heterocyclic reactions. Starting from (2, 5 – dimercapto -1, 3, 4-Thiadiazole) a variety of derivatives have been synthesis. Compound (1) was synthesized by the reaction of hydrazine hydrate with carbon disulphide in absolute ethanol. The compound (1) was reacted with 1, 2-dibromoethane in presence of alkali ethanol to give the compound (2). The compound (3) was formed from the reaction of compound (2) with hydrazine hydrate. Schiff base (4) was obtained by reacting of compound (3) with the compound (p-hydroxybenzaldehyde) in absolute ethanol. A variety of phenolic Schiff base (Methylolic, Etheric, and Epoxy) derivatives have been synthesized. Methylol

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Publication Date
Sun Jun 02 2013
Journal Name
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Synthesis and Characterization of some Mixed Ligand Complexes Containing (8-hydroxyquinoline) and (2 - picoline) with some Metal Ions
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Complexes of some metal ions ( Mn(I? ) , Co(??) , Ni(??) ,Cu (??) , Zn(I?) , Cd (??) , and Hg(??) ) with 8-hydroxyquinoline (Oxine) and 2- Picoline (2-pic ) have been synthesized and characterized on the basis of their FT-IR. and Uv-visible spectroscopy ,atomic absorption molar conductivity measurements and magnetic susceptibility ,from the results obtained the following general formula has been given for prepared complexes [M (oxine)2 (2-pic)2]where M = M(??) = Mn , Co , Ni , Cu , Zn , Cd , Hg(oxine)- = ionic ligand 8-hydroxyquinolin (oxinato)(2- pic) = 2- picoline

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Publication Date
Thu Dec 28 2023
Journal Name
Al–bahith Al–a'alami
Contents of the photo coverage of the Gulf Cup Championship on the pages of photojournalists on Instagram: An analytical study
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This study intends to examine the content of photo coverage related to the Gulf Cup (Gulf 25) as presented by Iraqi photojournalists on Instagram between 02/01/2023 and 27/01/2023. Utilizing content analysis, the two researchers employed the survey method to evaluate 674 digital photos posted by seven photojournalists specializing in this sporting event, which concluded with the Iraqi team's victory in the city of Basra.
Key findings from the study include:
1. The Category "Photos from the Gulf Cup matches" predominated, accounting for 291 instances or 42.794% of the total.
2. Photojournalists exhibited a keen interest in capturing images of attending fans and documenting celebrations surrounding th

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