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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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Investigations on the Impact of Using Elliptic Groynes on the Flow in Open Channels
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This paper presents a numerical simulation of the flow around elliptic groynes by using CFD ‎software. The flow was simulated in a flume with 4m long, 0.4m wide, ‎and 0.175m ‎high ‎‎with a constant bed slope. Moreover, the first Groyne placed at 1m from the flow ‎‎inlet with a ‎constant the Groyne height of 10cm and a 1cm thickness, and the ‎width of Groynes equals ‎7cm‎. A submergence ratio of the elliptic Groynes of 75% was assumed, corresponding to a discharge of ‎0.0057‎m3/sec. The CFD ‎model showed a good ability to simulate the flow ‎around ‎Groynes with ‎ good accuracy. The results of ‎CFD software showed that when using double elliptic Groy

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As the banking sector is a strong influence on the country's economic growth,The solid financial well-being of anybank does not mean only a guarantee for its investors, It is also important for both owners and workers and for theeconomy in all its joints.The elements of capital adequacy and quality of assets are important to the functioning of thebanking business.In this study, the research sample included four private banks. Quarterly data were used for the period(2011 - 2018).Moreover, data is also collected from articles, papers, the World Wide Web (the Internet) and specializedinternational journals.In this research, an effort was made to try to find out the effect of (the ratio of the capital owned todeposits on the value of the bank),

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The Effect of Brainstorming on audit Quality and its Reflection on Detecting the Risk of Fraud
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Brainstorming is one of the fundamental and necessary concepts for practising the auditing profession, as auditing standards encouraged the implementation of brainstorming sessions to reach reasonable assurance about the validity of the evidence and information obtained by the auditor to detect fraud, as the implementation of brainstorming sessions and the practice of professional suspicion during the audit process lead To increase the quality of auditing and thus raise the financial community's confidence in the auditing profession again after it was exposed to several crises that led to the financial community losing confidence in the auditing profession.

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The Effect of Brainstorming on audit Quality and its Reflection on Detecting the Risk of Fraud
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Brainstorming is one of the fundamental and necessary concepts for practicing the auditing profession, as auditing standards encouraged the implementation of brainstorming sessions to reach reasonable assurance about the validity of the evidence and information obtained by the auditor to detect fraud, as the implementation of brainstorming sessions and the practice of professional suspicion during the audit process lead to increase the quality of auditing and thus raise the financial community's confidence in the auditing profession again after it was exposed to several crises that led to the financial community losing confidence in the auditing profession.

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Contents of the photo coverage of the Gulf Cup Championship on the pages of photojournalists on Instagram: An analytical study
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This study intends to examine the content of photo coverage related to the Gulf Cup (Gulf 25) as presented by Iraqi photojournalists on Instagram between 02/01/2023 and 27/01/2023. Utilizing content analysis, the two researchers employed the survey method to evaluate 674 digital photos posted by seven photojournalists specializing in this sporting event, which concluded with the Iraqi team's victory in the city of Basra.
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Microwave assisted synthesis of new heterocyclic compounds: 1, 2, 3-triazoles and tetrazoles and study of their biological activity
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