Expansion of analytic functions in q-orthogonal polynomials

A classical result on the expansion of an analytic function in a series of Jacobi polynomials is extended to a class of q-orthogonal polynomials containing the fundamental Askey–Wilson polynomials and their special cases. The function to be expanded has to be analytic inside an ellipse in the complex plane with foci at ±1. Some examples of explicit expansions are discussed.      

On a problem of Rolewicz about Banach spaces that admit support sets

We construct an example of a nonseparable Banach space which does not admit a support set.² It is a consistent (and necessarily independent from the axioms of ZFC) example of a space C(K) of continuous functions on a compact Hausdorff K with the supremum norm. The construction depends on a construction of a Boolean algebra with some combinatorial properties. The space is also hereditarily Lindelöf in the weak topology but it doesn't have any nonseparable subspace nor any nonseparable quotient which is a C(K) space for K dispersed.     

Improved Bohr’s inequality for locally univalent harmonic mappings

We prove several improved versions of Bohr’s inequality for the harmonic mappings of the form $f=h+\overline{g}$, where $h$ is bounded by 1 and $\left|g^{\prime }\left(z\right)\right|\le \left|h^{\prime }\left(z\right)\right|$. The improvements are obtained along the lines of an earlier work of Kayumov and Ponnusamy, i.e. (Kayumov and Ponnusamy, 2018) for example a term related to the area of the image of the disk $D(0,r)$ under the mapping $f$ is considered. Our results are sharp. In addition, further improvements of the main results for certain special classes of harmonic mappings are provided.     

Geometrical forms of Abelès'' equations involving film thickness

Abelès' equations (in terms of ₁; ₂ and optical film thickness x) for all the eight optical functions (R_s; R_p; T_s; T_p; F_s; F_p; X_s and X_p) have been expressed in the form of equations representing standard geometrical three-dimensional figures. X_s and X_p are represented by plane surfaces whilst the remainder are skew cones. Translated into the n; k notation; X_s and X_p generate hyperboloids whilst the remainder produce paraboloids whose cross-sections are parts of figures of eight.     

Spin-adapted wave functions of the serber type and time reversal

A procedure is proposed for generating Serber-type spin eigenfunctions withM _s = 0. The procedure uses the time-reversal invariance of these functions to increase the efficiency and to reduce the storage requirements. Simplifications in calculating the matrix elements of an observable operator which follow from the use of the time-reversal symmetry are briefly discussed.     

生物素的全合成

本文综述国外学者在生物素合成方面所作的工作, 简述生物素的生化功能, 并指出在我国进行生物素合成开发研究的必要性和迫切性。     

Heat-capacity measurements and thermodynamic functions of crystalline adenine; revised thermodynamic properties of aqueous adenine

The standard molar heat capacity C^°_p,m of adenine(cr) has been measured using adiabatic calorimetry over the range 6<(T/K)<310 and the results used to derive thermodynamic functions for adenine(cr) at smoothed temperatures. At T=298.15 K, C^°_p,m=(142.67±0.29) J · K⁻¹ · mol⁻¹ and the third law entropy S^°_m=(145.62±0.29) J · K⁻¹ · mol⁻¹. The standard molar Gibbs free energy of formation Δ_fG^°_m at T=298.15 K for crystalline adenine was calculated, using the standard molar enthalpy of formation for the compound and entropies of the elements from the literature, and found to be Δ_fG^°_m=(301.4±1.0) kJ · mol⁻¹. The results were combined with solution calorimetry and solubility measurements from the literature to yield revised values for the standard molar thermodynamic properties of aqueous adenine at T=298.15 K: Δ_fG^°_m=(313.4±1.0) kJ · mol⁻¹, Δ_fH^°_m=(129.5±1.4) kJ · mol⁻¹, and S_m^°=(217.68±0.44) J · K⁻¹ · mol⁻¹.     

On the well-posedness of a two-phase minimization problem

We prove a series of results concerning the emptiness and non-emptiness of a certain set of Sobolev functions related to the well-posedness of a two-phase minimization problem, involving both the p(x)-norm and the infinity norm. The results, although interesting in their own right, hold the promise of a wider applicability since they can be relevant in the context of other problems where minimization of the p-energy in a part of the domain is coupled with the more local minimization of the L^∞-norm on another region.