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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
The Effect Of Thickness on The Optical Properties Of ZnS
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Zinc sulfide(ZnS) thin films of different thickness were deposited on corning glass with the substrate kept at room temperature and high vacuum using thermal evaporation technique.the film properties investigated include their absorbance/transmittance/reflectance spectra,band gap,refractive index,extinction coefficient,complex dielectric constant and thickness.The films were found to exhibt high transmittance(59-98%) ,low absorbance and low reflectance in the visible/near infrared region up to 900 nm..However, the absorbance of the films were found to be high in the ultra violet region with peak around 360 nm.The thickness(using optical interference fringes method) of various films thichness(100,200,300,and 400) nm.The band gap meas

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Publication Date
Fri Sep 01 2023
Journal Name
Journal Of Coloproctology
The Impact of Smoking on the Complications of Acute Appendicitis
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Abstract<p> Introduction The pathophysiology of appendicitis likely stems from obstruction of the appendiceal orifice leading to an increase in intraluminal and intramural pressure, resulting in small vessel occlusion and lymphatic stasis. Organ supplied by an end artery, such as the appendix, are more prone to the deleterious effects of arterial occlusion. The continuous cigarette smoking might be associated with a greater risk for developing atherosclerosis.</p><p> Objectives The aim is to evaluate the effect of cigarette smoking on the incidence of complications of acute appendicitis.</p><p> Methods Patients suffered from symptoms and signs suggesting a</p> ... Show More
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Publication Date
Wed Feb 19 2020
Journal Name
International Journal Of Development In Social Sciences And Humanities
The Effect of Context on the Comprehension of English Idioms
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DBN Rashid, International Journal of Development in Social Sciences and Humanities, 2020

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Publication Date
Thu Dec 01 2011
Journal Name
Journal Of Engineering
The Effect of Shear on the Properties of Rigid PVC
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The rheological and fusion behavior of polyvinyl chloride (PVC) compounds plays a dominant role in
the processing operations and in the development of physical properties in the processed material. A
comprehensive study was made in this work to evaluate the effect of shear and thermal history on stability, mechanical and rheological properties of rigid PVC compounds. Different samples of Rigid Poly vinyl chloride including dry blend powder, granules, and bottles molded from both were examined. A study was also made on recycled RPVC where 25% of reclaimed material was continuously blended with fresh dry blend and processed for 15 cycles. Results showed that compaction of the PVC material took place in the brabender plastograph at co

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Publication Date
Sat Dec 01 2012
Journal Name
Asian Journal Of Chemistry
Microwave assisted synthesis of new heterocyclic compounds: 1, 2, 3-triazoles and tetrazoles and study of their biological activity
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The work includes synthesis of 1,2,3-triazoles via click conditions and using the microwave irradiation starting from two synthesized azides: 2,3,4,6-tetra-O-acetyl-β-D-glucopyranosyl azide (5) and perfluorobutylethyl azide (10) and different terminal alkynes. It also includes microwave enhanced synthesis of tetrazoles via the reaction of two synthesized azides i.e., perfluorobutylethyl azide (10) and 1,5-diazidopentane (13) with benzoyl cyanide. Most of the prepared compounds have been characterized by: TLC, FT-IR, 1H NMR, 13C NMR, LC-MS and microelemental analysis

Publication Date
Sun Jan 01 2012
Journal Name
Chemistry And Materials Research
Synthesis ,characterization and antibacterial activity of mixed ligand complexes of some metals with 1-nitroso-2-naphthol and Lphenylalanine
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Publication Date
Wed Aug 10 2016
Journal Name
Al-nahrain
Synthesis and characterization of some Anti-bactrial Active Transition Metal Complexes of N-12-(2-phenyl hydrazinyl) phenyl [ benzothiohydrazide
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Publication Date
Mon May 20 2024
Journal Name
International Journal Of Diabetes In Developing Countries
Role of stanniocalcin-1 and proenkephalin-A as novel biomarkers in prediction of newly diagnosed type 2 diabetic patients
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Publication Date
Tue Sep 08 2020
Journal Name
Baghdad Science Journal
Silver Nanoparticles and their Role in Gene Expression of Motility Gene motB and Repression of AI-2-Controlled Gene
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Antibiotic resistance is the capability of the strains to resist or protect themselves from the effects of an antibiotic. Such a resistance towards the current antimicrobials leads to the search of novel antimicrobials. Nanotechnology has been promising in different field of science and among it is the use of nanoparticles as antibacterial agents. The gastrointestinal tract seems to be the primary reservoir of uropathogenic E.coli (UPEC) in humans. UPEC strains harbour the urinary tract and cause urinary tract infection. They cause serious ailments in terms of humans. They develop resistance and increase their virulence by forming biofilms. They also show a remarkable locomotory movement with the aid of autoinducer controlled ge

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Publication Date
Thu Mar 15 2018
Journal Name
Journal Of Baghdad College Of Dentistry
Assessment of Salivary Lactoferrin and pH Levels and Their Correlation with Gingivitis and Severity of Chronic Periodontitis (Part: 2)
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Background: Periodontal diseases are bacterial infections of the gingiva, bone and attachment fibers that support the teeth and hold them in the jaw. Lactoferrin is a multifunctional glycoprotein and it is the main component of neutrophil polymorphonuclear leukocytes that activated during inflammatory processes such as Periodontal diseases Aims of the study: Determine the salivary levels of Lactoferrin and pH and their correlations with clinical periodontal parameters(Plaque Index , Gingival Index , Bleeding on Probing , Probing Pocket Depth , and Clinical Attachment Level ) and the correlation between Lactoferrin with potential of hydrogen ion (PH) ,flow rate and α-amylase of study groups that consisted of patients had gingivitis and pa

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