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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
Thu May 28 2020
Journal Name
Iraqi Journal Of Science
The Association Between IL-2 Gene (RS2069763 (Single Nucleotide Polymorphism and Type 2 Diabetes Mellitus in Iraqi Patients
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This research attempts to find the association between single nucleotide polymorphism (SNP) of IL2+166 gene (rs2069763) and type 2 diabetes mellitus (T2DM) in a sample of Iraqi patients. A total of 44 patients and 55 apparently healthy volunteers were genotyped for the SNP using polymerase chain reaction test. Three genotypes (GG, GT, and TT) corresponding to two alleles (G and T) were found to have SNP. Both study groups’ genotypes had a good agreement for the analysis of Hardy-Weinberg Equilibrium. The results revealed increased frequencies between the observed and expected GG and TT genotypes and IL2+166 SNP T allele in T2DM  patients (40.9 vs. 40.0 %; OR = 1.04; 95% CI, 0.47 - 2.31), whereas the values in the control group were

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Publication Date
Sat Aug 21 2021
Journal Name
Open Access Macedonian Journal Of Medical Sciences
Immunohistochemical Expression of Retinoblastoma Gene Product and p53 Protein in Transitional Cell Carcinoma of the Urinary Bladder and its Relationship to Different Clinicopathological Parameters
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BACKGROUND: Carcinoma of urinary bladder is one of the most common malignancies worldwide and constitutes a major health problem. Multiple risk factors are associated with this tumor and its prognosis will depend on different clinicopathological parameters. Over expression of P53 protein and mutant Rb gene is associated with more aggressive clinical and histopathological features of the tumor such as advanced stage and higher grade. AIM: The immunohistochemical expression of Rb gene and P53 gene will be assessed through their protein products in transitional cell carcinoma (TCC) of the urinary bladder and then will be correlated with other well-known risk factors and prognostic parameters of bladder TCC, such as grading, tumor size, smoking

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Publication Date
Fri Apr 01 2016
Journal Name
Journal Of Economics And Administrative Sciences
Application Tools Lean Six Sigma Definition and measurement phases in new product development. In the Electrical Industries State Company. case Study.
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methodology six sigma Help  to reduce defects by solving problems effectively, and works Lean to reduce losses through the flow of the manufacturing process and when integrating these two methodologies (Lean and six sigma), the methodology of Lean six sigma will form the entrance to the organizers of the optimization process and increase the quality and reduce lead times and costs . by focusing on the needs of the customer. this process uses statistical tools and techniques to analyze and improve processes.

 We have conducted this research in the General Company for Electrical Industries and adopted its product (machine cooling water three taps) as a sample for research. In order to determine t

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Publication Date
Tue Feb 01 2022
Journal Name
International Journal Of Nanoscience
Study of the Transition State of SnO<sub>2</sub> Cluster with NO<sub>2</sub> Gas Molecule via Density Functional Theory
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Density functional theory (DFT) with B3LYP level and 6-311G[Formula: see text] basis sets for light atoms like N and O and SDD basis sets for heavy atoms like Sn is used to examine the interaction of tin dioxide nanocrystals with nitrogen dioxide as a function of temperature from 273[Formula: see text]K to 373[Formula: see text]K through a Gaussian 09W software program. Gibbs free energy, enthalpy, and entropy of activation and reaction are calculated. The situation of transition of SnO2 clusters toward nitrogen dioxide is investigated. According to the findings, the activation energy of SnO2 clusters with nitrogen dioxide increases as the temperature rises (in negative value). Gauss view 0

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Publication Date
Mon Feb 13 2023
Journal Name
Aip Conference Proceedings
composition,depiction,antibacterial,antioxidant,and cytotoxicity activities studies of a new nano-sized binuclear metal(||) schiff base complexes
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composition,depiction,antibacterial,antioxidant,and cytotoxicity activities studies of a new nano-sized binuclear metal(||) schiff base complexes

Publication Date
Mon Oct 01 2018
Journal Name
Journal Of Economics And Administrative Sciences
The effect of deficiencies in the standards and tools used for the management of the banking system in the emergence of the financial crisis by focusing on the Basel 1and 2
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The banking industry, as a result of the great challenges it faced, required continuous development of the principles of management, control and mechanisms used. The Basel Committee on Banking Supervision has played a leading role in legalizing many of these developments and has been able to contribute significantly to establishing a common framework for banking supervision, The central role in the various countries of the world is based on coordination between the various regulatory authorities and thinking about finding mechanisms to confront the risks faced by banks, recognizing the importance of the banking sector in the stability of domestic and international banking systems or the danger of this sector in the emergence of F

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Publication Date
Sun Sep 03 2017
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Kinetic and Mechanistic Studies for the Gas-phase Reaction of Ozone with 2, 3-Dimethyl-2-Butene and 1, 3-Butadiene
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The reactions of ozone with 2,3-Dimethyl-2-Butene (CH3)2C=C(CH3)2 and 1,3-Butadiene CH2=CHCH=CH2 have been investigated under atmospheric conditions at 298±3K in air using both relative and absolute rate techniques, and the measured rate coefficients are found to be in good agreement in both techniques used. The obtained results show the addition of ozone to the double bond in these compounds and how it acts as function of the methyl group substituent situated on the double bond. The yields of all the main products have been determined using FTIR and GC-FID and the product studies of these reactions establish a very good idea for the decomposition pathways for the primary formed compounds (ozonides) and give a good information for the effe

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Publication Date
Thu Nov 08 2018
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Chemistry – A European Journal
Magnetic Shielding, Aromaticity, Antiaromaticity and Bonding in the Low‐Lying Electronic States of S <sub>2</sub> N <sub>2</sub>
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Aromaticity, antiaromaticity and chemical bonding in the ground (S0), first singlet excited (S1) and lowest triplet (T1) electronic states of disulfur dinitride, S2N2, were investigated by analysing the isotropic magnetic shielding, σiso(r), in the space surrounding the molecule for each electronic state. The σiso(r) values were calculated by state-optimized CASSCF/cc-pVTZ wave functions with 22 electrons in 16 orbitals constructed from gauge-including atomic orbitals (GIAOs). The S1 and T1 electronic states were confirmed as 11Au and 13B3u, respectively, through linear response CC3/aug-cc-pVTZ calculations of the vertical excitation energies for eight singlet (S1–S8) and eight triplet (T1–T8) electronic states. The aromaticities of S

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Synthesis, Characterization of Poly Heterocyclic Compounds, and Effect on Cancer Cell (Hep-2) In vitro.
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A synthesis series of new heterocyclic derivatives (A2-A7) (pyrrole, pyridazine, oxazine and imidazol) derived from 4-acetyl-2,5-dichloro-1-(3,5-dinitrophenyl)-1H-pyrrole-3-carboxylate(A1) have been synthesised. Synthesis of compound (A2) by the reaction of starting material (A1) with hydroxyl amine hydrochloride in the presence of pyridine. Compound (A2) was reacted with hydrazine hydrate in dry benzene to give (A3) derivative. The compound )A3( deals with sodium nitrite to give diazonium salt, and the reaction diazonium salt with ethyl acetoacetate to produce compound (A4). To a mixture of compound (A4) and hydroxyl amine with sttired to yield (A5).Compound (A6) was prepared by reaction compound (A4) with thiosemicarbazide in presence

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Sat Oct 01 2022
Journal Name
The Egyptian Journal Of Hospital Medicine
Study on Viral Infection and Related Parameters in A Sample of Diabetes Mellitus Type 2
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