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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 Aug 16 2023
Journal Name
Journal Of Humanities And Social Sciences Research
The impact of the use of hydromassage rehabilitation exercises on the recovery of certain motor potentials of knee roughness sufferers
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The research included the introduction to the research and its importance as knee joint is an important joint in the human body that is prone to injury. One such injury is knee roughness injury that occurs as a result of the stress of the knee joint and age. The importance of examining the need for the use of rehabilitation exercises, especially in the watercourse system, is highlighted by the fact that the aquatic environment is one of the most important factors helping to alleviate pain and rehabilitate the knee joint and thereby improve the mobility of those with knee roughness. The problem of research is that rehabilitation exercises have been developed in the watercourse system on the basis of scientific bases with a repetitive and sys

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Publication Date
Wed Jun 17 2026
Journal Name
Journal Of Baghdad College Of Dentistry
In vivo immunohistochemical investigation of the effect of the topical application of growth hormone on the osseointegration of cpTi implant
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Background: Dental implants are a suitable option for the replacement of some or all missing teeth. The successful insertion of a biocompatible material into living tissue with little to no evidence of rejection has revolutionized medicine and dentistry. An increase in bone response was observed with local administration of growth hormone around dental implants. Growth hormone may act as a bone stimulant in the placement of endosseous dental implants and enhances osseointegration. The aim of the study was to evaluate immunohistochemically the effect of the topical application of growth hormone on the osseointegration of cpTi implant. Materials and Methods: Eighty titanium screw implants were inserted in the tibia of the forty adult rabbits.

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Publication Date
Mon Oct 01 2018
Journal Name
Journal Of Educational And Psychological Researches
The Effect of PDEODE Strategy on the Acquisition of Historical Concepts of First Class of Intermediate Students
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The research aims to find out the effect of PEDODE Strategy on the acquisition of historical concepts of First Intermediate Grade. To achieve this goal, the researcher has put forward this zero hypothesis: There is no statistically significant difference at the level of (05,0) between the mean of the students' marks in the experimental group who study the history subject using PEDODE strategy and that of the students' marks in the control group who study the subject according to the traditional method in the post-testing on the acquisition of historical concepts. The sample of the study consists of (62 female-students) of First Intermediate Grade in the Directorate of Baghdad Education/ Karkh2nd for the academic year 2016-2017. The sampl

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Publication Date
Tue Jan 01 2019
Journal Name
Journal Of Global Pharma Technology
Serum levels of cross-linked n-telopeptide of Type i collagen before and after non-surgical periodontal therapy in type 2 diabetic patients with chronic periodontitis
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Background: Diabetes mellitus is a major risk factor for chronic periodontitis (CP) and hyperglycemia has an important role in the enhancement of the severity of the periodontitis. It has been reported that the progression of CP causes shifting of the balance between bone formation and resorption toward osteoclastic resorption, and this will lead to the release of collagenous bone breakdown products into the local tissues and the systemic circulation. Cross-linked N-telopeptide of type I collagen (NTx) is the amino-terminal peptides of type I collagen which is released during the process of bone resorption. This study was conducted to determine the effects of nonsurgical periodontal therapy on serum level of NTx in type 2 diabetic patients

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Publication Date
Thu Aug 08 2019
Journal Name
Iop Conference Series: Materials Science And Engineering
A Study of poly4-oxo-4-(4-sulfamoylphenylamino) but-2-enoic acid as a good anti – Corrosion of 316L stainless steel in 0.2M hydrochloric acid solution.
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Publication Date
Tue Jun 01 2021
Journal Name
International Medical Journal
Surface analysis of the PEKK coating on the cp ti implant using laser technique
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Objective: Evaluation of the poly ether keton keton polymer (PEKK) coating material on the commercial pure titanium disks (CP Ti) with or without laser surface structuring. Design: In vitro experimental study of PEKK polymer coated material on the CP Ti disks with or without laser surface structuring. Materials and methods: coating the surface of the commercial pure titanium (CP Ti) disks with PEKK polymer was performed via using frictional mode CO2 laser, then the samples disks analyzed by using FESEM. Results: the FESEM reveal good adherence and distribution of the PEKK coated material over the CP Ti substrate by using the frictional mode CO2 laser at 2 watt and 6 ms pulse duration. Conclusion: the frictional mode CO2 laser considered an

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Publication Date
Sun Jan 01 2012
Journal Name
Turkish Journal Of Physics
The influence of confinements on the photon flux spectra in amorphous silicon quantum dots
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Publication Date
Sun Feb 03 2019
Journal Name
Journal Of The College Of Education For Women
Solution of the electromechanical machining problem using the collocation method based on Bernstein polynomials
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Publication Date
Thu Mar 15 2018
Journal Name
Journal Of Baghdad College Of Dentistry
The Impact of Mother-Infant Bonding on Periodontal Health Status in the Postpartum Period
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Background: Mother-infant bonding is an important psychological step postpartum and disturbed relationship may carry dramatic consequences as a psychological disorder which may affect the periodontal health of the mother. The aim of the present study was to assess the effect of the postpartum Mother-infant bonding on their periodontal condition. Materials and Methods: Mothers in the postpartum period with age range 20-35 years were subjected to postpartum Bonding Questionnaire (PBQ). Periodontal health status was assessed by measuring probing pocket depth and clinical attachment level. Results: The mean values of both probing pocket depth (PPD) and clinical attachment loss (CAL) were higher among disordered mothers than mothers with normal

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Publication Date
Mon Aug 01 2011
Journal Name
Journal Of Peptide Science
Studies on the antileishmanial properties of the antimicrobial peptides temporin A, B and 1Sa
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Given the paucity and toxicity of available drugs for leishmaniasis, coupled with the advent of drug resistance, the discovery of new therapies for this neglected tropical disease is recognised as being of the utmost urgency. As such antimicrobial peptides (AMPs) have been proposed as promising compounds against the causative Leishmania species, insect vector-borne protozoan parasites. Here the AMP temporins A, B and 1Sa have been synthesised and screened for activity against Leishmania mexicana insect stage promastigotes and mammalian stage amastigotes, a significant cause of human cutaneous disease. In contrast to previous studies with other species the activity of these AMPs against L. mexicana amastigotes was low. This suggests that ama

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