C. U. Shah University

This study presents detailed energy storage and biomedical application of hydrothermally synthesized MnSSe nanocomposites (NCs). Powder-XRD method was identified the coexistence of cubic MnS, MnSe2, and SeS phases with average crystallite size and dislocation density were estimated to be 16.72 nm and 1.11 & times; 10(16) m(-2) from Scherer formula and 15.78 and 0.16 & times; 10(16) from Williamson-Hall method, along with compressive microstrain of -1.1 & times; 10(-3) was observed. Furthermore, longitudinal optical phonon vibrations were analyzed by the Raman Spectroscopy. XPS and EDX analyses have been verified the materials purity as well as Mn2+, S2-, and Se2- oxidation states and near-stoichiometric composition. Additionally, FESEM and HRTEM micrographs were revealed prismatic and pyramidal morphologies, while SAED confirmed the polycrystallinity of MnSSe NCs. UV-Vis spectroscopy exhibited broad absorption, while direct bandgap was estimated to be 1.95 eV. Temperature dependent dielectric, conductivity, complex impedance/modulus analysis were investigated at temperature and frequency range of 148-423 K and 20 Hz - 1 MHz. The dielectric plots fitted with the Havriliak-Negami model exhibited a large dielectric strength (>10(3)), attributed to interfacial polarization arising from Mn-S/Se orbital hybridization. Coexistence of 3D Mott variable-range hopping and correlated barrier hopping mechanisms were confirmed from Jonscher's power-law followed conductivity. Moreover, complex impedance spectroscopy showed grain and grain-boundary contributions by modeling R-CPE equivalent circuits. Complex modulus spectroscopy was analyzed by R-Bergman and KWW model that confirmed thermally activated relaxation governed by localized charge motion and non-Debye type relaxation behavior. Antimicrobial analysis, including MIC measurements highlighted selective inhibition of Bacillus subtilis, attributed to gradual ion release and interfacial electron transfer disrupt bacterial metabolism.

We study Lipschitz algebras of holomorphic functions of the order k, 0 ≤ k ≤ ∞, and the exponent α, α ∈ (0, 1]. The Gel’fand theory and maximal ideal spaces of these algebras are discussed. Further, we solve the corona problem (1962) and Gleason’s problem (1964) for these algebras on certain bounded pseudoconvex/poly domains G in Cn (e.g., the ball and polydisc). As a welcome bonus, we affirmatively solve Fornæss and Øvrelid’s problem (1983) for holomorphic Hölder and Lipschitz spaces. In fact, we establish an equivalency between the two problems for these algebras. As an application, we establish the I.J. Schark’s theorem for Lipschitz algebras on these G’s. Indeed, we extend our recent work on Gleason’s problem, based on the functional analytic approach, as well as extend recent results of Clos for these algebras, and apply the usual Banach algebra method.

We use a combination of the classical dead-end space method and the interpolation (by bounded holomorphic functions) method to affirmatively solve Corona and Gleason problems for familiar Banach spaces of holomorphic functions such as (Besov-)Lipschitz spaces of holomorphic functions on bounded symmetric domains (including the unit ball and polydisc) in ℂ^n, holomorphic mean (Besov-)Lipschitz spaces, (Besov-)Hardy-Sobolev spaces and some special classes of holomorphic mixed-norm (Bergman) spaces (including weighted Bergman spaces) as well as some special classes of holomorphic (mean) Besov-Lipschitz spaces on the unit ball in ℂ^n. Indeed, we extend our recent work on both problems for holomorphic Lipschitz algebras.

We give an affirmative solution of the original Corona problem (1962) for certain bounded pseudoconvex domains or polydomains in ℂ^n, including the balls and polydiscs, by using the Gleason solvability property of such domains for H^∞(G). In fact, we establish an equivalency of the Corona problem and Gleason's problem (1964) in the theory of several complex variables. Indeed, we extend our recent works on both problems based on the functional analytic approaches, and extend recent work of Clos, as well as apply the usual Banach algebra method. As an application, we establish the I. J. Schark's theorem for these G's. We also obtain a welcome extension of a result of Krantz and Li (1995) for a bounded, strictly pseudoconvex domain in ℂ^n with C^3 boundary.

In 1952, Michael posed a question about functional continuity of commutative Fréchet algebras in his memoir, known as Michael's problem in the literature. We settle this in the affirmative along with its equivalent forms. Indeed, we continue our previous works to directly attack these problems. We show that the test case for this problem – the Fréchet algebra U of all entire functions on ℓ^∞ – is, in fact, a Fréchet algebra [[X]], if there exists a discontinuous character on U. An important tool is a topological version of the tensor (symmetric) algebra over a Banach space. The elementary, but crucial, idea is to express the test algebra U as a weighted Fréchet symmetric algebra over the Banach space ℓ^1. We summarize the effects of our affirmative solution in terms of important applications in automatic continuity theory, including affirmative answers to two significant, long-standing problems from 1966 and 1978.