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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
Thu Dec 01 2016
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Karbala Journal Of Medicine
Assessment Of Her-2/Neu Gene Amplification Status by Chromogenic in Situ Hybridization in Breast Cancer Patients with Equivocal 2+ Her-2/Neu Immunostaining and Its Relation to The Clinic Pathological Parameters
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background: human epidermal growth factor receptor-2 (her2/neu) is related to growth factor receptors with alkaline kinase activity and it is regarded as important prognostic and therapeutic factor that can depended on in breast cancer therapy. HER2/neu expression by immunohistochemistry (IHC) is submitted to a great in terob server inconsistency. Subsequently additional confirmatory tests for assessment of gene alterations and amplification status are needed for patients with early or metastatic breast cancer. In situ hybridization techniques and specifically Chromogenic in situ hybridization (CISH) was arise as a practical, cost-effective, and alternative to fluorescent in situ hybridization in testing for gene alterationAims of the study

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Publication Date
Sun Dec 15 2019
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Virtual Reality Technology and its Uses in Industrial Product Design: فلاح حسن هادي -صلاح نوري محمود
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The research (Virtual Reality Technology and its Uses in Industrial Product Design) is interested in the virtual reality technology used in the industrial product design and consequently knowing the functions achieved in the industrial product according to the data of that technology which participates in activating the mental and imaginary image of the user which show the parameters of the technical transformation of that product. The terms used in the research have been defined to guide the reader. The second chapter, the theoretical framework consisted of three sections the first is concerned with technology in the industrial design. The second is concerned with the virtual environment and the virtual reality. The thirds chapter consi

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Publication Date
Tue Oct 13 2026
Journal Name
Journal Of Baghdad College Of Dentistry
Assessment of tongue space area in a sample of Iraqi adults with class I dental and skeletal pattern
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Background: Lateral cephalometric radiography is commonly used as a standard tool in orthodontic assessment and treatment planning. This study aimed to determine the tongue and surrounding space area in a sample of Iraqi adults with class I dental and skeletal pattern. Materials and methods: The study included thirty healthy subjects (15 males and 15 females) with an age ranged between 23-34 years and class I dental and skeletal pattern with no history of any sleep related disorders. The assessed cephalometric measurement included length and height of the tongue and position of hyoid bone from cervical line. Descriptive statistics were obtained for the data. Genders difference was evaluated by independent sample t-test. Results: There wer

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Publication Date
Tue Jan 01 2019
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Al-dhad Book Store And Publishing
Calculus part 2
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This book includes four main chapters: 1. Indefinite Integral. 2. Methods of Integration. 3. Definite Integral. 4. Multiple Integral. In addition to many examples and exercises for the purpose of acquiring the student's ability to think correctly in solving mathematical questions.

Publication Date
Wed Jan 17 2018
Journal Name
Journal Of Applied Chemistry
Synthesis, Composition, Spectral, Geometryand AntibacterialApplicationsof MnII, NiII, CoII, CuII and HgIIcomplexes of N2O2-mixed donor Schiff Basewith 1,10-phenanthroline
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Publication Date
Mon Feb 13 2023
Journal Name
Journal: Aip Conference
Composition, Depiction, Antibacterial, Antioxidant, and Cytotoxicity activities studies of A New Nano-Sized Binuclear Metal (II) Schiff Base complexes
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Abstract In the current contribution, a novel binuclear nickel(II) and zinc(II) complexes were prepared from a hexadentate ligand prepared via condensation of 3,3'-Bipyridine-6,6'-dicarbaldehyde , 2-amino-5-chlorobenzaldehyde and 2-Aminophenol .The symmetric ligand (H2DTPE) and its metal complexes were illustrated utilizing various techniques of physicochemical containing magnetic moment, analytical analysis and spectroscopy of mass, IR, 13C and 1H NMR, TGA and UV-Vis. The particles of MO Nanoscale were created from the labeled complex applying the ways of pyrolysis and utilizing methods of XRD, FT-IR, and FE-SEM, that specified close compatibility with the typical pattern for nanoparticles of NiO, ZnO and appeared the reasonable size in

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Publication Date
Thu Feb 29 2024
Journal Name
International Journal Of Design &amp; Nature And Ecodynamics
Artificial Neural Network Assessment of Groundwater Quality for Agricultural Use in Babylon City: An Evaluation of Salinity and Ionic Composition
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Publication Date
Sun Sep 05 2010
Journal Name
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Composition operator induced by ?(z) = sz + t for which |s|?1, |t|<1 and |s|+|t|?1
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We study in this paper the composition operator that is induced by ?(z) = sz + t. We give a characterization of the adjoint of composiotion operators generated by self-maps of the unit ball of form ?(z) = sz + t for which |s|?1, |t|<1 and |s|+|t|?1. In fact we prove that the adjoint is a product of toeplitz operators and composition operator. Also, we have studied the compactness of C? and give some other partial results.

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Publication Date
Wed May 19 2010
Journal Name
Journal Of College Of Science, University Of Babylon
H-Point Standard Addition Method for Simultaneous Determination of Cimetidine and Erythromycin Ethylsuccinate Drugs Using Bromothymol Blue as a Chromogenic Complexing Agent
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The H-Point Standard Addition Method (H-PSAM) has been applied for spectrophotometric simultaneous determination of Cimetidine and Erythromycin ethylsuccinate using Bromothymol Blue (BTB) as a chromogenic complexing agent in a buffer solution at pH 5.5.

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Fri Aug 15 2025
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Synthesis, Characterization, Biological Activity and Antioxidant Studies of New Metal Ion Complexes Based on (E)-4-((benzo[d]thiazol-2-ylimino)methyl)-1,5-dimethyl-2-phenylpyrazolidin-3-one
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