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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
Hard-soft interacation of 2-mercaptobenzonathol with some metal ions
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Was appointed acid steady disintegration of organic EkandThe results proved that organic Allicand acting and Konnh solid baseBy Tgrav Pearson has possible account Maamat hardness and softness of organic Ekand

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Publication Date
Sun Sep 01 2013
Journal Name
Baghdad Science Journal
Synthesis of Some Heterocyclic Compounds Derived from 2-Mercapto Benzoxazole
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New series of 2-mecapto benzoxazole derivatives (1-20) incorporated into fused to different nitrogen and suphur containing heterocyclic were prepared from 2-meracpto benzoxazole, when treated with hydrazine hydrate to afford 2-hydrazino benzoxazol (1). Compound (1) converted to a variety of pyridazinone andphthalazinone derivatives (2-4) by reaction with different carboxylic anhydride. Also, reaction of (1) with phenyl isothiocyanate and ethyl chloro acetate afforded 3-phenyl-1,3-thiazolidin-2,4-dione-2-(benzoxazole-2-yl-hydrazone) (6). Azomethines (7-10) were prepared through reaction of (1) with aromatic aldehyde, then (7, 8) converted to thaizolidinone derivatives (11, 12). Treatment of (1) with active methylene compounds afforded deriva

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Publication Date
Sun Mar 01 2009
Journal Name
Baghdad Science Journal
Synthesis of Some New 1,2,4-Triazoles Derived from 2-Mercaptobenzimidazole
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New 1,2,4-triazole derivatives of 2-mercaptobenzimidazole (MB) are reported. Ethyl (benzimidazole-2-yl thio) acetate (1) has been prepared by condensing 2-mercaptobenzimidazole with ethylchloroacetate. The ester (1) on reacting with hydrazine hydrate gave the corresponding acetohydrazide(2)which was reacted separately with phenylisocyanate and phenylisothiocyanate, followed by ring closure in an alkaline medium giving 3-[(benzimidazole-2-yl thio) methyl]-4-phenyl-1,2,4-triazole-5-ol and 3-[(benzimidazole-2-yl thio) methyl]-4-phenyl-1,2,4-triazole-5-thiol respectively (6,7). Reaction of acetohydrazide (2) with CS2 and ethanol/KOH, gave dithiocarbazate salt (8). Cyclization of (8) with hydrazine hydrate gave 3-[(benzimi

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Publication Date
Tue Jun 14 2011
Journal Name
Journal Of Education
Synthesis and Characterization Complexes of 2- Thiotolylurea with Matel Salts.
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Complexes of some metal ions with 2-thiotolylurea were prepared in ethanolic medium using (1:1) (Metal : Ligand) ratio yielded series of neutral complexes as the general formula [M(L)Cl2]. The prepared complexes were identified by atomic absorption FT.IR, UV-Visble spectra, molar conductivity and magnetic properties. From the above data the tetrahedral structure was suggested for all complexes.

Publication Date
Sun Apr 30 2023
Journal Name
Al-kindy College Medical Journal
SARS-CoV-2 and other Coronaviruses: A matter of variations
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Since the appearance of COVID-19 disease as an epidemic and pandemic disease, many studies are performed to uncover the genetic nature of the newly discovered coronavirus with unique clinical features. The last three human coronavirus outbreaks, SARS-CoV, MERS-CoV and SARS-CoV-2 are caused by Beta-Coronaviruses. Horizontal genetic materials transfer was proven from one coronavirus to the other coronavirus of non-human origin like infectious bronchitis virus (IBV) of avian. Horizontal genetic materials transfer was also from non-corona viruses like astroviruses and equine rhinovirus (ERV-2) or from coronavirus-unrelated viruses, like influenza virus type C. However, SARS-CoV-2 is identical to SARS-CoV and MERS-CoV. Interestingly, Wuhan ci

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Publication Date
Fri Jan 01 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-1H-pyrazol-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 octahedral.

Publication Date
Thu May 17 2018
Journal Name
Ibn Al-haitham Journal For Pure And Applied Science
Synthesis, Characterization and Study of Biological Activity of Some New Schiff Bases, 1, 3-Oxazepine and Tetrazole Derived from 2, 2 di thiophenyl Acetic Acid
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In this study new derivatives of Schiff bases 5-8, 1, 3-oxazepine 9-16 and tetrazoles 17-19 have been synthesized from the new starting material 1 which has synthesized the reaction of one mole of dichloro acetic acid and two moles of thiophenol, the esters 2-3 were synthesized from the reaction of compound 1 with methanol or ethanol respectively in the presence of H2SO4 as catalyst then 2, 2-dithiophenylaceto Hydrazide 4 were synthesized from the reaction of 2 or 3 with hydrazine hydrate 80%, Schiff bases 5-8 were synthesized from the reaction of 4 with appropriate aldehyde or ketone. Treatment of Schiff bases with maleic and phathalic anhydride in dry benzene to give 1, 3-oxazepen derivatives 9-16 and with sodium azide in tetrahydrofuran

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Publication Date
Fri Aug 26 2022
Journal Name
Journal Of Contemporary Medical Sciences
Measurement of the serum level of Leucine-rich alpha-2-glycoprotein-1 in hospitalized Iraqi COVID-19 Patients
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Objective: The study aimed to assess Leucine-rich alpha-2-glycoprotein-1 biomarker serum level in hospitalized COVID-19 patients. Methods: The case control study from multi-centers in Baghdad included 45 adult patients (19 females and 26 males) with COVID-19, diagnosed with a positive real-time reverse transcription polymerase chain reaction and excluded negative RT-PCR for COVID-19 and comorbidity conditions. Second group, was 43 control (20 females and 23 males). Results: This study found a decrease Leucine-rich alpha-2-glycoprotein-1 biomarker serum level in these patients and a significant difference in D. dimer, neutrophil count, lymphocyte count, and the neutrophil-lymphocyte ratio between the patients and controls at a P valu

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Publication Date
Mon Aug 31 2020
Journal Name
Bmc Oral Health
The success of using 2% lidocaine in pain removal during extraction of mandibular premolars: a prospective clinical study
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Abstract<sec><title>Background

The purpose of this study was to evaluate the anesthetic effectiveness of a buccal infiltration technique combined with local massage (using 2% lidocaine) in the extraction of mandibular premolars to be utilized as an alternative to the conventional inferior alveolar nerve block.

Methods

Patients eligible included any subject with a clinical indication for tooth extraction of the mandibular 1st or 2nd premolars. All patients were anesthetized buccally by local infiltration technique followed by an external pressure applied for 1 min directly over the injection area. In each case, another local

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Publication Date
Sat Dec 15 2018
Journal Name
Journal Of Baghdad College Of Dentistry
Comparative evaluation of the radiographic density of GuttaFlow® 2 in different obturation techniques using Spiral Computed Tomography
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Background: Ideal root canal obturation depends on many factors; one of them is good sealing of root canal without pores. The aim of this study was to determine the radiographic density of GuttaFlow® 2 with different obturation techniques using spiral computed tomography. Materials and Methods: Forty palatal roots of permanent maxillary first molar were used in this study. Following working length determination, root canal was prepared using rotary PROTAPER universal system. They were randomly divided into four groups of 10 roots each, the groups are Conventional lateral condensation with Apexit Plus sealer, Conventional lateral condensation with GuttaFlow® 2 as a sealer, Soft Core Regular with GuttaFlow® 2 as a sealer and singl

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