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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
Tue Dec 26 2017
Journal Name
International Journal Of Science And Research (ijsr)
Investigating the Effect of Using Waste Glass on the Properties of Asphalt Concrete Wearing Course Mixture
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The reuse or recycling of waste materials in different aspects of life is served the objective of sustainability and be beneficial to society. In recent years, a wide variety of waste materials were used in pavement construction. One of these materials is glass that generally produces in large quantities and crushed glass can be considered feasible alternative source of aggregate for asphalt mixture production. This study focused on examining the asphalt mixture properties of wearing course using crushed glass as fine aggregates. Fine crushed glass with various percentages by total weight retained on sieve 2.36 mm, 0.3 mm and 0.075 mm was used in the study. The results indicate that mixes containing crushed glass had lower Marshall stabilit

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Publication Date
Thu Nov 08 2018
Journal Name
Iraqi National Journal Of Nursing Specialties
The effect of disinfectants on the surface quality of irreversible hydrocolloid impression material and gypsum cast
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Objective: To evaluate the changes in the surface quality of irreversible hydrocolloid impression material
hydrogum following disinfection with 0.525% sodium hypochlorite, 0.2% Chlorehexidine Gluconate, and 4%
Povidone Iodine.
Methodology: Forty specimens of alginate impression materials hydrogum were fabricated according to the
ISO 1563 and were divided into four groups according to the method of solution dipping: group 1: Dip in
0.525% sodium hypochlorite, group 2: Dip in 0.2% chlorhexidine gluconate, Group 3: Dip in 4% Povidone Iodine,
Group 4: No treatment with any solution (control group). Then the specimens were poured in type II stone.
Surface detail was determined using a stainless steel block in accordance w

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Publication Date
Wed Apr 30 2025
Journal Name
Iraqi Journal Of Science
The Influence of Fear on the Dynamics of Harvested Prey-Predator Model with Intra-Specific Competition
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The influence of fear on the dynamics of harvested prey-predator model with intra-specific competition is suggested and studied, where the fear effect from the predation causes decreases of growth rate of prey.  We suppose that the predator attacks the prey under the Holling type IV functional response. he existence of the solution is investigated and the bounded-ness of the solution is studied too. In addition, the dynamical behavior of the system is established locally and globally. Furthermore, the persistence conditions are investigated. Finally, numerical analysis of the system is carried out.

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Publication Date
Fri Oct 01 2010
Journal Name
Iraqi Journal Of Physics
The effect of TiO2 addition on the thermal conductivity of Polymethylmetha acrylate, Polycarbonate and Polystyrene Polymers
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Thermal conductivity measurement was done for specimens of Polystyrene/ titanium dioxide, Polycarbonate/ titanium dioxide and Polymethylmetha acrylate/ titanium dioxide composites for weight ratio of 1.9/ 0.1 and 1.8/ 0.2 wt% for different thickness of the samples. The experimental results show that the thermal conductivity is increased with the increasing of thickness of layers and with the weight ratio of TiO2

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Publication Date
Sat Dec 01 2018
Journal Name
Journal Of Accounting And Financial Studies ( Jafs )
Measuring the impact of the financial value of human resources on investor decisions: (An Empirical Study)
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The aim of the research is to clarify the measurement of the impact regarding financial value of human resources on investors' decisions by considering that the human element is one of the assets of the company. Therefore, a set of criteria must be available to determine the applicability of these standards in the human resource because it has an effective role in the success for the company. Is to measure the value of human resources in a financial format according to the first two methods depends on the value and the second depends on the cost.

In order to achieve the objectives of the study, a questionnaire was designed to survey the views of a number of employees of the General Company for Leather Industries in order to arriv

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Publication Date
Fri Sep 15 2023
Journal Name
Al-academy
Contemporary Iraqi arts techniques and their reflection on the products of the students of Art Education
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The techniques of contemporary Iraqi painting and their reflection on the productions of students of art education is an important subject in the field of painting at the theoretical and practical levels in academic study, whether theoretical or practical. Al-Iraqi is one of the arts with historical roots and a distinguished position among other artistic genres. Painting has received a sufficient level of development through the use of various contemporary techniques to advance it for the better.
The methodological framework included the problem of research and the need for it, and then the importance of research came in shedding light on the techniques of contemporary Iraqi painting, and the impact of these techniques on the producti

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Publication Date
Thu Dec 01 2022
Journal Name
Iraqi Journal Of Physics
Study the Effect of Dielectric Barrier Discharge (DBD) Plasma on the Decomposition of Volatile Organic Compounds
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Recently, research has focused on non-thermal plasma (NTP) technologies as a way to remove volatile organic compounds from the air stream, due to its distinctive qualities, which include a quick reaction at room temperature. In this work, the properties of the plasma generated by the dielectric barrier discharge (DBD) system and by a glass insulator were studied. Plasma was generated at different voltages (3, 4, 6, 7, 8 kV ) with a fixed distance between the electrodes of 5 mm, and a constant argon gas flow rate of (2.5) I/min. DBD plasma emission spectra were recorded for each voltage. The Boltzmann plot method was used to calculate the electron temperature in the plasma ( ), and the Stark expansion method was used to calculate the elec

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Publication Date
Sun Jun 20 2021
Journal Name
Bulletin Of The Iraq Natural History Museum (p-issn: 1017-8678 , E-issn: 2311-9799)
ON THE ECOLOGY AND SPECIES DIVERSITY OF THE FRESHWATER GASTROPODS OF SPRINGS IN ANDIJAN REGION, UZBEKISTAN
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This study examines the species composition, biodiversity, zoogeography, and ecology of freshwater gastropods of 12 springs in Andijan region of Uzbekistan. The study used generally accepted malacological, faunistic, ecological, analytical, and statistical methods. As a result of research in the springs, 14 species of freshwater gastropods belonging to 2 subclasses, 5 families, and 10 genera were recorded. 7 of them are endemic to Central Asia. When indicators of biodiversity of mollusks were analyzed according to the Shannon index, it was found that the highest value was recorded in the springs besides the hills. According to the biotope of distribution and bioecological features, they were divided into cryophilic, phytophilic, pelophil

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Publication Date
Wed Oct 18 2023
Journal Name
Iraqi National Journal Of Nursing Specialties
The Effect of a Training Program on the Diabetic Patients Utilizing Skills of Insulin Injection Technique
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Objectives: The study aims to evaluate patients’ performance toward insulin injection after training program to
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The sample of the study consisted of (n =30) males and females

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Publication Date
Sat Feb 12 2022
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The Effect of Different Curing Temperatures on the Properties of Geopolymer Reinforced with Micro Steel Fibers
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In this study, geopolymer mortar was designed in various experimental combinations employing 1% micro steel fibers and was subjected to different temperatures, according to the prior works of other researchers. The geopolymer mortar was developed using a variety of sustainable material proportions (fly ash and slag) to examine the influence of fibers on its strength. The fly ash weight percentage was 50%, 60%, and 70% by slag weight to study its effect on the geopolymer mortar's properties. The optimal ratio produced the most significant results when mixed at a 50:50 ratio of fly ash and slag with 1% micro steel fibers at curing temperature 240oC for 4 hours through two days. The compressive strength of the geopolymer mortar increas

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