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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
Fri Jan 01 2021
Journal Name
Aip Conference Proceedings
Plasma levels and diagnostic utility of MMP-2 and TIMP-2 in the diagnosis of Iraqi women with breast tumor: A comparative study with Ca 15-3
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Publication Date
Fri Jan 01 2021
Journal Name
Aip Conference Proceedings
Plasma levels and diagnostic utility of MMP-2 and TIMP-2 in the diagnosis of Iraqi women with breast tumor: A comparative study with Ca 15-3
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Publication Date
Sun May 12 2013
Journal Name
International Journal Of Pharmacy And Pharmaceutical Sciences
PLEIOTROPIC EFFECTS OF ATORVASTATIN ON PRO-INFLAMMATORY CYTOKINES IN NEWLY DIAGNOSED IRAQI PATIENTS WITH TYPE 2 DIABETES MELLITUS
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Objective: Atorvastatin therapy is now recommended for reduction of cardiovascular risk in type 2 diabetic patients (T2DM), based on convincing evidence of reductions in mortality and vascular events in major clinical outcome trials. The aim is to evaluate the effects of atorvastatin on proinflammatory markers (TNF-α, IL-6), HbA1c andleptin in obese patients with type 2 diabetes. Methods: Sixty fivenewly diagnosed T2DM patients were randomly allocated into 2 groups; group I treated with metformin only; in group II atorvastatin was added with metformin. Twenty healthy subjects were enrolled as control group. While maintaining their usual eating habits, fasting blood samples were collected at baseline and after 12 weeks of treatment. Results

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Publication Date
Sun Jun 07 2015
Journal Name
Baghdad Science Journal
On The Nearby-Tip Strain Investigation and Failure-Propability Evaluation for Impacted Thin Plates Using the 2-Random-Variables Multi-Canonical-Based Joint Propability Distributions
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The study of the validity and probability of failure in solids and structures is highly considered as one of the most incredibly-highlighted study fields in many science and engineering applications, the design analysts must therefore seek to investigate the points where the failing strains may be occurred, the probabilities of which these strains can cause the existing cracks to propagate through the fractured medium considered, and thereafter the solutions by which the analysts can adopt the approachable techniques to reduce/arrest these propagating cracks.In the present study a theoretical investigation upon simply-supported thin plates having surface cracks within their structure is to be accomplished, and the applied impact load to the

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Publication Date
Sat Jan 01 2022
Journal Name
1st Samarra International Conference For Pure And Applied Sciences (sicps2021): Sicps2021
Consequences for the groups 𝒮𝓛(2,𝒰), 𝒰 = 23 and 29
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Publication Date
Wed Mar 15 2023
Journal Name
Journal Of Baghdad College Of Dentistry
Different methods of canine retraction- Part 2
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Background: This review aims to discuss various canine retraction techniques using frictionless mechanics. Methods: Between 1930 and February 2022, searches were conducted about various canine retraction techniques using fixed orthodontic appliances in various databases, including PubMed Central, Science Direct, Wiley Online Library, the Cochrane Library, Textbooks, Google Scholar, Research Gate, and manual searching. Results: After removing the duplicate articles, publications that described how to use archwires to perform canine retraction with the archwires were included. Conclusions: The pros and cons of various canine retraction techniques using archwires were thoroughly discussed. T-loop is the preferred spring of all because of it

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Publication Date
Tue Aug 05 2025
Journal Name
Journal Of Baghdad College Of Dentistry
Expression of matrix metalloproteinase-2 in the extracellular matrix of osseointegrated and diseased implants
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Background: Recently with improvement of dental implantology science, osseointegrated implants show a considerable durability, however; failures are not completely avoidable. Matrix metalloproteinase-2 (MMP-2) expression is disturbed in many pathological conditions such as peri-implantitis and periodontitis. This study was carried out to investigate the tissue expression of MMP-2 in the extracellular matrix of osseointegrated and diseased implants. Subjects and methods: Gingival biopsies were collected from six patients having osseointegrated or working implants and twenty with diseased or non osseointegrated implants and (6) controls having no implants. In situ hybridization technique was used to analyze the changes in immunoreactivity of

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Publication Date
Fri Apr 01 2011
Journal Name
The Iraqi J. Agric. Sci.
THE RELATIVE CONSTANT TO ESTIMATE ̅ 2 OF 3–WAY AND DOUBLE CROSSES OF MAIZE
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A field experiment was conducted at the field of the Dept. of Field Crop Sci. / College of Agriculture / University of Baghdad . The objective was to determine the values of relative constant of three – way and double crosses of maize . Ten inbreds were used and crossed during spring and fall seasons of 2009 to produce three - way and double crosses , and ten hybrids were taken from each group . The ten hybrids were grown and selfed during spring 2010 to produce 2 seed . Three way and double crosses were sown with their parents and 2 seed during fall 2010 in RCBD with four replicates . Leaf area , total dry matter , row/ear , grain/ear , grain weight and grain weight/plant of hybrids , parents and 2 plants were taken . Results showed that

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Publication Date
Sat Oct 01 2022
Journal Name
Digest Journal Of Nanomaterials And Biostructures
Preparation and study effect of vacuum annealing on structure and optical properties of AgCuInSe<inf>2</inf> thin film
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Publication Date
Wed Sep 01 2021
Journal Name
Baghdad Science Journal
Effect of Testosterone Enanthate Modeling of Polycystic Ovary on Liver Irs-2 mRNA Expression in Rats: A Brief Report
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There are many animal models for polycystic ovary (PCO); using exogenous testosterone enanthate is one of the methods of induction of these models. However, induction of insulin resistance should also be studied in the modeling technics. Therefore, the present study aims to investigate the expression of insulin receptor substrate (Irs)-2 mRNA in the liver tissue of rat PCO model. Nineteen Wistar rats were divided into three groups; (1) PCO modeling group (N =7) received daily 1.0 mg/100g testosterone enanthate solved in olive oil along with free access dextrose water 5%, (2) vehicle group (N =6), which handled like the PCO group, but did not receive testosterone enanthate, (3) control group (N =6) with standard care. Al

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