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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 Sep 25 2020
Journal Name
International Journal Of Drug Delivery Technology
Synthesis, Antibacterial, and Molecular Docking Study of Novel 2-Chloro-8-Methoxy-3-Aryl-[1,3] Benzoxazine Derivatives using Vilsmeier Reagent
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Reducing of ethyl 4-((2-hydroxy-3-methoxybenzylidene)amino)benzoate (1) afford ethyl 4-((2-hydroxy-3-methoxybenzyl)amino)benzoate (2). Reaction of this compound with Vilsmeier reagent affords novel 2-chloro-[1,3] benzoxazine ring (3). The corresponding acid hydrazide of compound 3 was synthesized from reaction of compound (3) with hydrazine hydrate. Newly series of hydrazones (5a–i) were synthesized from reaction of acid hydrazide with various aryl aldehydes. Antibacterial activity of the hydrazones was secerned utilizing gram-negative and gram-positive bacteria. Compound (5b) and (5c) exhibited significant antibacterial ability against both gram-negative and gram-positive bacteria, while the compounds (5a) showed mild antibacteri

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Publication Date
Thu Jul 02 2020
Journal Name
International Journal Of Drug Delivery Technology
Synthesis, Antibacterial and Molecular Docking Study of Novel 2-Chloro-8-Methoxy-3-Aryl-[1,3] Benzoxazine Derivatives Using Vilsmeier Reagen
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Reducing of ethyl 4-((2-hydroxy-3-methoxybenzylidene)amino)benzoate (1) afford ethyl 4-((2-hydroxy-3-methoxybenzyl) amino)benzoate (2). Reaction of this compound with Vilsmeier reagent affords novel 2-chloro-[1,3] benzoxazine ring (3). The corresponding acid hydrazide of compound 3 was synthesized from reaction of compound (3) with hydrazine hydrate. Newly series of hydrazones(5a–i) were synthesized from reaction of acid hydrazide with various aryl aldehydes. Antibacterial activity of the hydrazones wassecerned utilizing gram-negative and gram-positive bacteria. Compound (5b) and (5c) exhibited significant antibacterial ability against both gram-negative and gram-positive bacteria, while the compounds(5a) showed mild antibacterial activit

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Publication Date
Sun Mar 04 2012
Journal Name
Baghdad Science Journal
Synthesis and Spectral Studies of ZnII,CdII and HgII Complexes with 5-(2-Benzoic acid azo)-8-hydroxy quinoline Ligand
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Coupling reaction of 2-amino benzoic acid with 8-hydroxy quinoline gave bidentate azo ligand. The prepared ligand has been identified by Microelemental Analysis,1HNMR,FT-IR and UV-Vis spectroscopic techniques. Treatment of the prepared ligand with the following metal ions (ZnII,CdII and HgII) in aqueous ethanol with a 1:2 M:L ratio and at optimum pH, yielded a series of neutral complexes of the general formula [M(L)2]. The prepared complexes have been characterized by using flame atomic absorption, (C.H.N) Analysis, FT-IR and UV-Vis spectroscopic methods as well as 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 ra

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Publication Date
Thu Oct 25 2018
Journal Name
Al-nahrain Journal Of Science
Synthesis of some New 3-Substituted Heterocyclic Compounds Containing Bridgehead Nitrogen from 2-Amino Pyridine with Study Their Biological Activity
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Publication Date
Wed May 24 2017
Journal Name
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Direct and Indirect Spectrophotometric Determination of Paracetamol in Pharmaceutical Preparations By Oxidative Coupling With p-Amino-2-hydroxy Sodium Benzoate
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New simple and sensitive spectrophotometric methods for the determination of paracetamol in aqueous medium were developed. The first method is based on coupling of paracetamol with p-amino-2-hydroxy sodium benzoate (AHB) in the presence of sodium periodate, as oxidizing agent, to form a brownish-orange compound which shows a λmax at 470 nm. The molar absorptivity (εmax) of the colored product was found to be (3371) l. mole1. cm-1 and Sandel’s index 0.0449 μg. cm-2. The method follows Beer’s law in the concentration range of 12.5-500.0 μg of paracetamol in a final volume of 25 ml (0.5-20.0) μg. ml-1 with relative standard deviation percent (RSD%) ranged between 0.26-4.71% and accuracy, expressed by recovery percent, 95-106% for five

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Tue Oct 30 2018
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Catalytic Hydrogenation of <i>p</i>-Chloronitrobenzene to <i>p</i>-Chloroaniline Mediated by γ-Mo<sub>2</sub>N
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Promoting the production of industrially important aromatic chloroamines over transition-metal nitrides catalysts has emerged as a prominent theme in catalysis. This contribution provides an insight into the reduction mechanism of p-chloronitrobenzene (p-CNB) to p-chloroaniline (p-CAN) over the γ-Mo2N(111) surface by means of density functional theory calculations. The adsorption energies of various molecularly adsorbed modes of p-CNB were computed. Our findings display that, p-CNB prefers to be adsorbed over two distinct adsorption sites, namely, Mo-hollow face-centered cubic (fcc) and N-hollow hexagonal close-packed (hcp) sites with adsorption energies of −32.1 and −38.5 kcal/mol, respectively. We establish that the activation of nit

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Sun Jul 08 2018
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Association of Insulin-Like Growth Factor 2 Apa 1 A820G (rs680) Polymorphism with Thyroid Dysfunction in a Sample Iraqi Patien
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Synthesis and Spectral Studies of ZnII,CdII and HgII Complexes with 5-(2-Benzoic acid azo)-8-hydroxy quinoline Ligand
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Coupling reaction of 2-amino benzoic acid with 8-hydroxy quinoline gave bidentate azo ligand. The prepared ligand has been identified by Microelemental Analysis,1HNMR,FT-IR and UV-Vis spectroscopic techniques. Treatment of the prepared ligand with the following metal ions (ZnII,CdII and HgII) in aqueous ethanol with a 1:2 M:L ratio and at optimum pH, yielded a series of neutral complexes of the general formula [M(L)2]. The prepared complexes have been characterized by using flame atomic absorption, (C.H.N) Analysis, FT-IR and UV-Vis spectroscopic methods as well as 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

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Treatment of Acne Vulgaris with 5-Alpha Avocuta Cream 2% in Comparison with Tretinion Cream 0.025%(Single Blind Comparative Study)
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Sun Sep 01 2013
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Synthesis and Theoretical Study of 4-(2-methyl-4-oxoquinazoline-3(4H)-yl) Benzoic acid with Some Transition Metal Ions
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New complexes of the type [ML2(H2O)2] ,[FeL2(H2O)Cl] and [VOL2] were M=Co(II),Ni(II) and Cu(II) ,L=4-(2-methyl-4-oxoquinazoline-3(4H)-yl) benzoic acid were synthesized and characterized by element analysis, magnetic susceptibility ,molar conductance ,FT-IR and UV-visible. The studies indicate that the L acts as doubly monodentate bridge for metal ions and form mononuclear complexes. The complexes are found to be octahedral except V(IV) complex is square pyrimde shape . The structural geometries of compounds were also suggested in gas phase by theoretical treatments, using Hyper chem-6 program for the molecular mechanics and semi-empirical calculations, addition heat of formation(?Hf ?) and binding energy (?Eb)for the free ligan

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