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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
Wed Oct 01 2008
Journal Name
Saudi Medical Journal
Topical therapy of acne vulgaris using 2% tea lotion in comparison with 5% zinc sulphate solution.
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KE Sharquie, AA Noaimi, MM Al-Salih, Saudi Medical Journal, 2008 - Cited by 56

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Publication Date
Wed Jun 10 2015
Journal Name
Journal Of The College Of Basic Education
Synthesis, characterization and antimicrobial studies of 2-(4-antipyrine azo)-4-nitroaniline with some metal ions.
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Transition metal complexes of Co(II), Ni(II), Cu(II), and Zn(II) with 2-(4-antipyrine azo)-4-nitroaniline derived from 4-aminoantipyrine and 4-nitroaniline were synthesized. Characterization of these compounds has been done on the basis of elemental analysis, electronic data, FT-IR, UV-Vis and 1HNMR, as well as magnetic susceptibility and conductivity measurements. The nature of the complexes formed were studied following the mole ratio and continuous variation methods, Beer's law obeyed over a concentration range (1×10-4 - 3×10-4 M). High molar absorbtivity of the complex solutions were observed. From the analytical data, the stoichiometry of the complexes has been found to be 1:2 (metal:ligand). On the basis of physicochemical data octa

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Publication Date
Sun Jan 01 2023
Journal Name
Iraqi Journal Of Hematology
Evaluation of angiopoietin-2 level in patients with multiple myeloma at presentation and in remission state
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The study aimed to assess the level of ANG‑2 in MM patients at diagnosis and in remission state and elaborate on its correlation with interleukin‑6 (IL‑6) and beta‑2 microglobulin (B2M) levels. Sixty MM patients; 20 newly diagnosed (ND), and 40 patients in remission were included. Twenty healthy individuals were included as a control group. Plasma levels of ANG‑2, B2M, and IL‑6 were tested by enzyme‑lin ked immunosorbent assay. There are significant statistical differences between ND patients and those in remission in hemoglobin, neutrophil count, blood urea, serum creatinine, glomerular filtration rate, B2M, IL6, and ANG‑2 (P = 0.001, 0.033, 0.005, 0.001, 0.001, 0.001, 0.004, and 0.001, respectively). ANG‑2 showed signifi

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Publication Date
Wed Oct 01 2008
Journal Name
Journal
Topical therapy of acne vulgaris using 2% tea lotion in comparison with 5% zinc sulphate solution
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S Khalifa E, N Adil A, AS Mazin M…, 2008

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Publication Date
Thu Oct 13 2016
Journal Name
Brazilian Journal Of Oral Sciences
Expression of TIMP-2 in HPV-16 infected oral squamous cell carcinoma in patients in Iraq
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Aim: To determine the expression of tissue inhibitors of metalloproteinases (TIMP-2) in oral squamous cell carcinoma (OSCC) and the difference in its expression level between positive and negative HPV-16 (human papilloma virus- 16) OSCC patients. Methods: This study was conducted on 33 biopsies obtained from patients with OSCC and 10 normal oral mucosa as controls. In situ hybridization (ISH) was used to investigate the presence of HPV-16, while immunohistochemistry (IHC) was used to estimate the expression level of TIMP-2. Results: The TIMP-2 was expressed in 27 (81.8%) of OSCC sections with no significant difference between its expression level in HPV-16 positive and HPV-16 negative OSCC cases (p=0.058). TIMP-2 was found to be hig

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Publication Date
Thu Oct 01 2020
Journal Name
Iraqi Journal Of Applied Physics
Spectral and Electrical Characteristics of Nanostructured NiO/TiO 2 Heterojunction Fabricated by DC Reactive Magnetron Sputtering
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In this work, p-n junctions were fabricated from highly-pure nanostructured NiO and TiO2 thin films deposited on glass substrates by dc reactive magnetron sputtering technique. The structural characterization showed that the prepared multilayer NiO/TiO2 thin film structures were highly pure as no traces for other compounds than NiO and TiO2 were observed. It was found that the absorption of NiO-on-TiO2 structure is higher than that of the TiO2-on-NiO. Also, the NiO/TiO2 heterojunctions exhibit typical electrical characteristics, higher ideality factor and better spectral responsivity when compared to those fabricated from the same materials by the same technique and with larger particle size and lower structural purity.

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Publication Date
Tue Mar 30 2021
Journal Name
Baghdad Science Journal
Synthesis, Characterization and Biological Activity Evaluation of Some Pyrazoles, Thiazoles and Oxazoles Derived from 2-Mercaptoaniline
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    Synthesis of 2-mercaptobenzothiazole (A1) is performed from the reaction of  o-aminothiophenol and carbon disulfide CS2 in ethanol under basic condition. Compound (A1)  is reacted with chloro acetyl chloride to give compound (A2). Hydrazide acid compound (A3) is obtained from the reaction of compound (A2) with  hydrazine hydrate in ethanol under reflux in the presence of glacial acetic acid .The reaction of hydrazide acid compound (A3) with ethyl acetoacetate gives pyrazole compound (A4). The new hydrazone  compound (A5) was prepared from the reaction of compound (A3) with  benzaldehyde. Reaction of compound

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Publication Date
Wed Apr 01 2020
Journal Name
Indian Journal Of Forensic Medicine & Toxicology
Evaluation of some inflammatory cytokines and glycated hemoglobin in uncontrolled type 2 diabetes mellitus with nephropathy
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Background: Chronic hyperglycemia causes diabetic nephropathy(DN), which is a typical microvascular complication of type 2 diabetes mellitus. The pathogenesis of DN is not fully understanding. The inflammation may possess a significant role in the progression of DN in diabetic patients. Method: The study accomplished at teaching laboratories of medical city, Baghdad, Iraq. It was included 50uncontrolled diabetic type 2 patients with nephropathy, age range (40-78) years and 42 controlled diabetics type 2 without nephropathy, age range (35 - 52) years as a control group. The participants divided in to two groups according to HbA1c measurement which is described as follows: < 7.5% of HbA1c describes controlled diabetes, and > 9% of HbA1c

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Publication Date
Sun Oct 21 2018
Journal Name
Al-kindy College Medical Journal
Biochemical Risk Determinants of Osteoporosis in Overweight and Obese Postmenopausal Women with Type 2 Diabetes Mellitus
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Background: Several studies suggested that skeletal system is adversely affected by diabetes and is associated with increased risk of osteoporosis and fragility fractures  

Objectives: The study was a case-control study that designed to assess the level of bone turnover markers (BTMs) among patients with type 2 diabetes mellitus (T2DM) and to investigate the effect of body weight and diabetic control on the level of bone turnover

Type of the study: Cross- sectional study.

Methods: The present study included 100 postmenopausal women with type 2 diabetes mellitus. Sixty-six non-diabetic postmenopausal women were enrolled as a control. Fasting b

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Publication Date
Sun Jul 01 2018
Journal Name
Journal Of Global Pharma Technology
Synthesis, Characterization and Investigation Liquid Crystalline, Properties of 1, 2, 3-triazole Derivatives via Cycloaddition Reaction
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The new of compounds synthesized by sequence reactions starting from a reaction of 3-phenylenediamine or 4-phenylenediamine with chloroacetyl chloride to produce the compounds [I]a,b, then the compounds[I]a,b reacted with sodium azide to yield compounds[II]a,b that reacted 1,3-dipolarcycloaddition reaction with acrylic acid to give compounds [III]a,b these compounds reacted with methanol led to ester compounds[IV]a,b then reacted with hydrazine to give acid hydrazide [V]a,b . Finally compounds [V]a,b reacted with aromatic aldehydes to product shiff bases derivatives. The compounds characterized by mp. , IR, 1HNMR in addition to mass spectroscopy for some of them the liquid crystals properties were studied by using polarized optical microsco

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