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We prove four theorems on the uniqueness of non linear differential polynomials sharing one value which improve a result of Yang and Hua, and supplements some results of Lahiri, Xu and Qiu and Banerjee.
In this paper, we have studied on the uniqueness problems of meromorphic functions with its linear c-shift operator in the light of partial sharing. Our two results improve and generalize two very recent results of Noulorvang-Pham [Bull. Korean Math. Soc. 57 (2020), no. 5, 1083-1094] in some sense. In addition, our other results have improved and generalized a series of results due to Lü-Lü [Comput. Methods Funct. Theo. 17 (2017), no. 3, 395-403], Zhen [J. Contemp. Math. Anal. 54 (2019), no. 5, 296-301] and Banerjee-Bhattacharyya [Adv. Differ. Equ. 509 (2019), 1-23]. We have exhibited a number of examples to show that some conditions used in our results are essential.
In this paper, we have studied on the uniqueness problems of meromorphic functions with its linear $c$-shift operator in the light of partial sharing. Our two results improve and generalize two very recent results of Noulorvang-Pham [Bull. Korean Math. Soc. 57 (2020), no. 5, 1083--1094] in some sense. In addition, our other results have improved and generalized a series of results due to L\"u-L\"u [Comput. Methods Funct. Theo. 17 (2017), no. 3, 395--403], Zhen [J. Contemp. Math. Anal. 54 (2019), no. 5, 296--301] and Banerjee-Bhattacharyya [Adv. Differ. Equ. 509 (2019), 1--23]. We have exhibited a number of examples to show that some conditions used in our results are essential.
<P><B>Abstract</B></P> <P>CuAlO<SUB>2</SUB> is a technologically important material having diverse applications, including superior thermoelectric properties. Its unique crystallographic structure manifests an anisotropic environment for the charge carriers and phonons, which is considered to be the reason for the enhancement of the thermopower. To exploit this novel property, a controlled <I>sol-gel</I> deposition technique is adopted to realize highly <I>c</I>-axis-oriented growth of CuAlO<SUB>2</SUB> thin film on Si and glass substrates. Thermoelectric measurements are performed in such a way that the carriers are confined along the <I>a-b</I> plane of the nanocrystal, which is parallel to the substrate. This allows a two-dimensional confinement of the charge carriers, leading to enhanced thermoelectric properties. Additionally, the temperature-dependent electrical characterizations depict two different charge-transport regimes with a cross-over at 360K. The low-temperature region corresponds to the mobility-activated small-polaron conduction and the high-temperature region belongs to the semiconductor-type carrier-density-activated conduction. Calculation of polaron activation energy from low-temperature regime indicates considerable influence of band carriers (hole) on the polaronic levels, due to which the above-mentioned transition is manifested. Calculations of activation and Fermi energy from high-temperature regime reveal a deep acceptor level and shallow Fermi level, which is typical of a non-degenerate semiconductor with acceptors not fully ionized at room temperature.</P>
<P>The postsynthetic modification strategy is adopted to demonstrate for the first time the syntheses of catalytically active chiral MOPMs from a preassambled achiral framework, MIL-101, by attaching L-proline-derived chiral catalytic units to the open metal coordination sites of the host framework. Various characterization techniques (including PXRD, TGA, IR, and N(2) absorption measurements) indicated that the chiral units are successfully incorporated into MIL-101, keeping the parent framework intact. The new chiral MOPMs show remarkable catalytic activities in asymmetric aldol reactions (yield up to 90% and ee up to 80%). It is interesting to note that these heterogeneous catalysts show much higher enantioselectivity than the corresponding chiral catalytic units as homogeneous catalysts. This study demonstrates a simple and efficient route for the generation of catalytically active chiral MOPMs. A variety of chiral catalytic units can be, in principle, incorporated into chemically robust achiral MOPMs with large pores by postmodification and the resulting chiral MOPMs may find useful applications in catalytic asymmetric transformations.</P>
<P>While various mineralizing peptides have been applied to grow metal nanoparticles on bionanotube templates, the semiconductor nanoparticle growth on nanotubes has not extensively been explored yet. In this paper, various semiconductor nanocrystals were grown on the bionanotubes surfaces with controlled sizes. When three synthetic peptides, which recognize and selectively bind Ge, Ti, and Cu ions, respectively, were incorporated on template bionanotube surfaces, highly crystalline and monodisperse Ge, TiO2, and Cu2S nanocrystals were grown on the tube surfaces. The sizes of these nanocrystals could be tuned as a function of pH, and larger semiconductor nanocrystals were grown as the pH of growth solutions was increased. All of these nanocrystals from smaller sizes to larger sizes had the same crystallinity. This peptide-controlled nanocrystal growth technique will be very useful to prepare semiconductor nanowires as building blocks for future microelectronics, whose band gaps can be tuned by the sizes of coated semiconductor nanoparticles via their quantum confinement effect. The novelty of this approach in the electronic device fabrication is that the semiconductor nanocrystal size control can be achieved by controlling peptide configurations via pH change, and this control may tune electronic structures and band gaps of the resulting semiconductor nanowires.</P>
<P>We define and study the notion of numerical equivalence on algebraic cobordism cycles. We prove that algebraic cobordism modulo numerical equivalence is a finitely generated module over the Lazard ring, and it reproduces the Chow group modulo numerical equivalence. We show this theory defines an oriented Borel-Moore homology theory on schemes and oriented cohomology theory on smooth varieties. We compare it with homological equivalence and smash-equivalence for cobordism cycles. For the former, we show that homological equivalence on algebraic cobordism is strictly finer than numerical equivalence, answering negatively the integral cobordism analogue of the standard conjecture (D). For the latter, using Kimura finiteness on cobordism motives, we partially resolve the cobordism analogue of a conjecture by Voevodsky on rational smash-equivalence and numerical equivalence. (C) 2015 Elsevier B.V. All rights reserved.</P>