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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
Sun Mar 07 2010
Journal Name
Baghdad Science Journal
The role of some types of erythrocytes on the growth of Entamoeba histolytica trophozoite in vitro
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The parasite E.histolytica was first isolated from a stool sample, and then cultivated and maintained in vitro using Locke-egg medium (LEM) and Liver infusion agar medium (LIAM) . Then, the effect of some types of erythrocytes (human and sheep), on the growth and activity of the parasite in the two culture media was investigated. The parasite was able to ingest and lysis erythrocytes of human and sheep that were supplemented to the culture media and such manipulation was able to augment the reproduction rate of the cultivated E. histolytica, however, such consequence was media- and concentration-dependent. The reproduction rate was significantly increased (66.0, 57.5 and 58.6%, respectively) in LEM medium containing human erythrocytes ty

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Publication Date
Wed Mar 01 2023
Journal Name
Al-khwarizmi Engineering Journal
Study of the Effect of Magnetic Abrasive Finishing on the Material Removal of AA1100 Aluminum Alloy
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This study evaluates the performance of magnetic abrasive finishing (MAF) of aluminum alloy in terms of achieving materials removal (MR). A vertical milling machine is used to perform the finishing process using a developed MAF unit that consists of an inductor made out of a 150 mm long and 20 mm diameter iron core wound with 1500 turns and 0.5 mm copper wire. The commutator and magnetic pole are attached at the top and bottom of the inductor, respectively. The required current is supplied using a DC power supply. The South Pole workpiece is a 100×50×3 mm3 plate of AA 1100 aluminum alloy, whereas the magnetic pole represented the North Pole. Pole rotational speed, applied current, and abrasive finishing time was selected as

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Publication Date
Sun Sep 06 2015
Journal Name
Baghdad Science Journal
The Effect of nano particles of TiO2-Al2O3 on the Mechanical properties of epoxy Hybrid nanocomposites
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Preparation of epoxy/ TiO2 and epoxy/ Al2O3 nanocomposites is studed and investigated in this paper. The nano composites are processed by different nano fillers concentrations (0, 0.01, 0.02 ,0.03, 0.04 ,0.05 ,0.07 and 0.1 wt%). The particles sized of TiO2,Al2O3 are about 20–50 nm.Epoxy resin and nano composites containing different shape nano fillers of (TiO2:Al2O3 composites),are shear mixing with ratio 1 to 1,with different nano hybrid fillers concentrations( 0.025 ,0.0 5 ,0.15 ,0.2, and 0.25 wt%) to Preparation of epoxy/ TiO2- Al2O3 hybrid composites. The mechanical properties of nanocomposites such as bending ,wearing, and fatigue are investigated as mechanical properties.

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Publication Date
Fri Nov 01 2013
Journal Name
International Journal Of Advanced Research
The effects of alkaloid extracted of Solanum nigrum leaves on the biological performance of Dialerodus citri
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Publication Date
Sun Jun 06 2010
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Baghdad Science Journal
The effect of Sunn bug Eurygaster intgriceps infection of wheat on the Capability of bread making
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The present study aims to identifying the effect of Eurygaster integriceps infection on the quality features of the local wheat varity IBA 99. The result revealed significant decrease in the percentage of protein of infected wheat at (4, 5)% compared to sound wheat values for the sound wheat were (10.9%) whereas those for the infected wheat were(9.0, 8.4 )% also the result revealed significant increased in the percentage of ash in the infected wheat flour. The result showed significant decrease in the percentage of wet and dry gluten in the infected wheat by sun bug at the rate of (2, 3, 4, 5)% compared to sound wheat value (27, 9% and 25, 23, 22, 22, 20%) for wet gluten and (8.2, 7.8, 7, 7.2, 6%) for dry glutin . The value of sedimentati

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Publication Date
Mon May 01 2023
Journal Name
Materials Today: Proceedings
An experimental study of the effects of matrix acidising on the petrophysical characteristics of carbonate formation
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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
Mon Oct 13 2025
Journal Name
Journal Of Economics And Administrative Sciences
The effect of creating knowledge according to the model (Nonaka &amp; Takeuchi, 1995) on organizational ambidexterity: A study on a sample of Iraqi private banks
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الغرض - تعتمد هذه الدراسة على المنهج الوصفي التحليلي من خلال جمع البيانات اللازمة وتحليلها، كون هذا المنهج يركز على استطلاع الآراء لعينة البحث وتوجهاتها ، وتهدف إلى تطوير نموذج يدرس العلاقة بين خلق المعرفة والبراعة التنظيمية في المصارف الخاصة العراقية والتحقق من صحته تجريبياً. التصميم / المنهجية / المدخل- تم إجراء مسح عبر استمارة استبيان لجمع البيانات من عينة من (113) مدير من مصارف تجارية خاصة بالإضافة إلى ذلك ا

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Publication Date
Wed Sep 11 2024
Journal Name
Lecture Notes In Civil Engineering
An Image Processing Algorithm to Address the Problem of Stains Merge on Water Sensitive Papers and Its Impact on the Evaluation of Spray Quality Indicators
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There are many techniques that can be used to estimate the spray quality traits such as the spray coverage, droplet density, droplet count, and droplet diameter. One of the most common techniques is to use water sensitive papers (WSP) as a spray collector on field conditions and analyzing them using several software. However, possible merger of some droplets could occur after they deposit on WSP, and this could affect the accuracy of the results. In this research, image processing technique was used for better estimation of the spray traits, and to overcome the problem of droplet merger. The droplets were classified as non-merged and merged droplets based on their roundness, then the merged droplets were separated based on the average non-m

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