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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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Sun Jun 01 2025
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SARS-CoV-2 history, diagnosis, genetic diversity and treatment: a comprehensive review
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SARS-CoV-2 history, diagnosis, genetic diversity and treatment: a comprehensive review
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This article comprehensively examines the history, diagnosis, genetics, diversity, and treatment of SARS-CoV-2. It details the emergence of coronaviruses over the past 50 years, including the coronavirus from 2019 and its subsequent mutations, along with updated information about this virus. This review explains the development and nomenclature of coronaviruses, their cellular invasion through glycoprotein spikes binding to ACE-2 receptors, and the mechanism of cell entry via endocytosis. Diagnosis methods for COVID-19, including nucleic acid amplification, serology, and imaging techniques like chest X-ray and CT scan tests, are discussed. Treatment approaches for COVID-19 are outlined, emphasizing healthcare, antiviral medications like Rem

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Awareness Regarding Diabetes Mellitus and Its' Complications in Type 2 Diabetic Patients
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Background: Various studies conducted in many parts of the world suggest that there is lack of public awareness and knowledge of various aspects related to diabetes. With proper education, awareness, earlier detection and better care, many complications and co-morbidities can be reduced in diabetic population.Objectives: to evaluate the level of awareness of diabetes mellitus type 2 patients regarding their disease and its' complications.Methods: Cross – sectional survey was conducted during November and December 2011, in the Medical centers of Al Baladiat, Mustansyria and Zuafranya, including 145 type 2 diabetic patients (58.6 % males, 41.4% females) who were subjected to self–structured questionnaires regarding different aspects of

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Pryparation, characterization and biological activity of new derivatives of 2-biphenyl-3-aminomethylimidazo(1,2-a)pyrimidine
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New Fourteen compounds were synthesized in four steps. The first step included synthesis of 2-biphenyl fused ring of imidazo(1,2- a)pyrimidine from the reaction of 2-aminopyrimidine and biphenyl phenacyl bromide . The second step was introduced aldehyde group from the reaction of 2-biphenyl fused rings of imidazo(1,2-a)pyrimidine with POCl3 in presence of DMF and CHCl3. 3-Carbaladehyde derivatives of fused imidazo/pyrimidine was reacted with different aromatic amines to afford new Schiff bases. These new 3- imines derivatives was reduced by using sodiumborohydride to yield another new 3-aminomethyl-2-biphenyl imidazo (1,2-a)pyrimidine derivatives in moderate yield .Some new prepared compounds were identified by melting point, FT- IR , 13C-

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Preparation of some azo compounds by diazotization and coupling of 2- amino -5 – thiol -1,3,4- thiadizaole
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2- amino -5- thiol-1,3,4- thiadiazole (S1) was prepared by cyclic locking of thiosemicarbazide in the presence of anhydrous sodium carbonate and CS2. diazotization of (S1) compound gave diazonium salt (S2) that reacts with different activated aromatic compounds to get the following azo compounds ,2 [(4- aminophenyl) diazenyl ] 1,3,4- thiazdiazole-5- thiol (S3) ,2-[4-amino- 1-naphthyl diazenyl] -1,3,4 – thiazdiazole-5-thiol (S4) , 3-amino-4-[(5- mercapto -1,3,4- thiadiazole -2-yl) diazenyl ] phenol(S5) ,1-[(5-mercapto-1,3,4-thiadiazole-2-yl) diazenyl] -2-naphthol (S6) , 5-{[4-(dimethylamino) phenyl] diazenyl}-1,3,4-thiadiazole-2- thiol(S7) ,5-{[4-(diethylamino) phenyl] diazenyl}-1,3,4- thiadiazole-2- thiol(S8) ,2- amino-5-[(5-mercapto-1,3

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ABSTRACT : A new ligand [ 2- (3-acetylthioureido)-3-phenylpropanoic acid (APA) is synthesized by reaction of acetyl isothiocyanate with phenylalanine (1:1). It is characterized by micro elemental analysis (C.H.N.S.), FT-IR, (UV-Vis) and 1H and 13CNMR spectra. Some metals ions complexes of this ligand were prepared and characterized by FT-IR, UV-Visible spectra, conductivity measurements, magnetic susceptibility and atomic absorption. From results obtained, the following formula [M(APA)2] where M2+ = Mn, Co, Ni, Cu, Zn, Cd and Hg, the proposed molecular structure for these complexes as tetrahedral geometry, except copper complex is has square planer geometry.

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Comparison of the structural motifs and packing arrangements of six novel derivatives and one polymorph of 2-(1-phenyl-1H-1,2,3-triazol-4-yl)pyridine
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The crystal structures of a new polymorph and seven new derivatives of 2-(1-phenyl-1H-1,2,3-triazol-4-yl)pyridine have been characterized and examined along with three structures from the literature to identify trends in their intermolecular contact patterns and packing arrangements in order to develop an insight into the crystallization behaviour of this class of compound. Seven unique C-H...X contacts were identified in the structures and three of these are present in four or more structures, indicating that these are reliable supramolecular synthons. Analysis of the packing arrangements of the molecules using XPac identified two closely related supramolecular constructs that are present in eight of the 11 structures; in all cases, the st

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