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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
Sun Jun 05 2022
Journal Name
Egyptian Journal Of Chemistry
Synthesis, Characterization, and Biological Activity of New Metal Ions Complexes with Schiff Base Derived from 2-Acetylthiophene and Isatin-3-Hydrazone
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This study describes the preparation of a new bidentate Schiff base derived from the condensation of Isatin-3-hydrazone with 2-acetylthiophene and the preparation of new series of complexes with a good yield. The prepared ligand was characterized by IR, UV-Vis, C.H.N.S elemental analysis, 1H and 13C NMR, LC-Mass spectroscopy, and physical measurements. Its complexes were analyzed by C.H.N.S elemental analyses, UV-Vis., FTIR, NMR, LC-Mass Spectra, atomic absorption spectroscopy, magnetic susceptibility, and conductivity measurements The results from spectroscopy and measurement studies showed that the ligand coordinated to the metal ion as a bidentate ligand via oxygen and nitrogen, forming an octahedral geometry around it. In vitro antimicr

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Tue May 06 2014
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Gastroprotective Activity of Ethyl-4-[(3,5-di-tert-butyl-2-hydroxybenzylidene) Amino]benzoate against Ethanol-Induced Gastric Mucosal Ulcer in Rats
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The study was carried out to determine the cytotoxic, antioxidant and gastro-protective effect of ethyl-4-[(3,5-di-tert-butyl-2-hydroxybenzylid ene)amino] benzoate (ETHAB) in rats.

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Publication Date
Sun Dec 01 2013
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Theoretical Study of Charge Transfer in Styryl Thiazilo Quinoxaline Dyes STQ-1, STQ-2,and STQ-3 in Organic Media System
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Charge transfer in styryl dyes STQ-1, STQ-2,and STQ-3 with organic media system has been studied theoretically depending on the Franck- Condon rule and continuum dielectric model . The reorientation energies (eV) were evaluated theoretically depending on dipole momentum, dielectric constant , and refrective index n. The rate constant of charge transfer has been calculated depending on the reorientation energy (eV) ,effective free energy , potential height barrier , and coupling coefficient . A matlap program has been written to calculated the rate constant of charge transfer and other parameter. The results of calculations show that STQ-2 dye is more reaction for charge transfer compare with STQ-1 and STQ-3 dyes

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Publication Date
Wed Jan 01 2020
Journal Name
Indian Journal Of Forensic Medicine And Toxicology
Continuous Flow Injection Analysis,Turbidmetric and Photometric Determination of Methyldopa Using a New Long Distance Chasing Photometer (NAG-ADF-300-2)
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A specific, sensitive and new simple method was used for the determination of methyldopa in pure and pharmaceutical formulations by using continuous flow injection analysis. This method is based on formation of ion pair compound between methyldopa and potassium hexacyanoferrate in acidic medium to obtain a yellow precipitate complex using long distance chasing photometer (NAG-ADF-300-2). The linear range for calibration graph was 0.05-35 mmol/L for cell A and 0.05-25 mmol/L for cell B, and LOD 1.4292 µg /200 µL for both cells with correlation coefficient (r) 0.9981 for cell A and 0.9994 for cell B, RSD% was lower than 0.5 % for n=8 for. The results were compared with classical method UV-Spectrophotometric at λ max=280 nm and turbi

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Publication Date
Tue Apr 01 2025
Journal Name
Nigerian Postgraduate Medical Journal
Pain and Satisfaction among Type 2 Diabetic Patients: Effects of Coughing, ShotBlocker and Vibration during Insulin Injection: A Randomised Controlled Trial
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Background:

Diabetes mellitus is a chronic illness that commonly leads to progressive and incapacitating of patients’ condition over the past 20 years.

Aim:

The aim of the study was to evaluate the effects of the coughing technique, ShotBlockerTo evaluate the effects of the coughing technique, ShotBlocker and vibration device on pain intensity and patient satisfaction during subcutaneous (SC) insulin injections in hospitalised adults with Type 2 diabetes mellitus (T2DM).

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Publication Date
Fri Jul 21 2023
Journal Name
Journal Of Engineering
The Effect of Laminated Layers on the Flutter Speed of Composite Wing
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The paper presents an investigation to the flutter speed of composite wing for different ply orientation. Structurally the composite wing was idealized as a composite beam load carrying structure. Theodorsen’s expression was used to get the 2- dimension unsteady lifting force and pitching moment in the limit of incompressible flow and subsonic speed which were integrated over the wing span. A free vibration analysis was first carried out to get the natural frequencies and mode shapes .The velocity-damping (V-g) method was used to calculate the flutter speed and the flutter frequency. A wing of unmanned aerial vehicle was manufactured from woven glass and polyester resin where the flutter speed was calculated experimentally by the wind

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Publication Date
Sun Sep 05 2010
Journal Name
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The Effects of ? – Rays on The Optical Constants of ZnS Thin Films
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ZnS thin films were grown onto glass substrates by flash evaporation technique, the effects of ? – rays on the optical constants of ZnS these films were studied. It was found that ? – rays affected all the parameters under investigation.

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Publication Date
Wed Dec 11 2024
Journal Name
Tikrit Journal Of Engineering Sciences
The Effects of Fibers on the Properties of Local Hot Asphalt Mixtures
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Conventional flexible pavements are released to different types of failure in the initial phases of their service life due to high traffic density, high speeds, heavy loads, and harsh climates. To eliminate pavement damage and failure early, the present search investigates the impact of adding glass, steel, and basalt fibers in the asphalt mixtures. Also, the study evaluates these materials characteristics compared to the mixtures without fibers. The Marshall test and tensile strength ratio test (TSR) were utilized to evaluate the asphalt mixture's performance. A set of specimens were produced by incorporating glass fiber (GF), steel fiber (SF), and basalt fiber (BF) at (0.10%, 0.15%, 0.20%), (0.25%, 0.35%, 0.45%), and (0.15%, 0.35%

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Publication Date
Fri Jan 01 2016
Journal Name
International Journal Of Advanced Research
ON STABILITY AND THE BOUNDEDNESS OF THE ZERO SOLUTION OF FIFTH ORDER
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This paper is illustrates the sufficient conditions of the uniformly asymptotically stable and the bounded of the zero solution of fifth order nonlinear differential equation with a variable delay τ(t)

Publication Date
Tue Oct 12 2021
Journal Name
Engineering, Technology &amp; Applied Science Research
The Effect of Nanomaterials on the Properties of Limestone Dust Green Concrete
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Portland cement is considered the most involved product in environmental pollution. It is responsible for about 10% of global CO2 emissions [1]. Limestone dust is a by-product of limestone plants and it is produced in thousands of tons annually as waste material. To fulfill sustainability requirements, concrete production is recommended to reduce Portland cement usage with the use of alternative or waste materials. The production of sustainable high strength concrete by using nanomaterials is one of the aims of this study. Limestone dust in 12, 16, and 20% by weight of cement replaced cement in this study. The study was divided into two parts: the first was devoted to the investigation of the best percentage of replacement of waste

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