Hilbert’s Nullstellensatz for analytic trigonometric polynomials

This paper proves such a new Hilbert’s Nullstellensatz for analytic trigonometric polynomials that if ${\left\{f_j\right\}}_{j=1}^{n\ge 2}$ are analytic trigonometric polynomials without common zero in the finite complex plane $?$ then there are analytic trigonometric polynomials ${\left\{g_j\right\}}_{j=1}^{n\ge 2}$ obeying ${\sum }_{j=1}^{n\ge 2}{f}_j{g}_j=1$ in $?$, thereby not only strengthening Helmer’s Principal Ideal Theorem for entire functions, but also finding an intrinsic path from Hilbert’s Nullstellensatz for analytic polynomials to Pythagoras’ Identity on $?$.     

The Green polynomials via vertex operators

An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood function. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas are given for Green's polynomials associated with upper partitions of length ≤3 and the diagonal lengths ≤3; (3) a Murnaghan-Nakayama type formula for the Green polynomial is obtained; and (4) an iterative formula is derived for the bitrace of the finite general linear group G and the Iwahori-Hecke algebra of type A on the permutation module of G by its Borel subgroup.     

Influence of acceptor tethering on the performance of nonlinear optical properties for pyrene-based materials with A-π-D-π-D architecture

In this study, we designed a series of pyrene-based donor-π-donor-π-acceptor compounds (HPTC1-HPTC7) by structural tailoring the reference compound (HPTC) using acceptor units. Nonlinear optical (NLO) properties, frontier molecular orbitals (FMOs), natural bonding orbital (NBO), transition density matric (TDM) analysis, and absorption spectra of reference and proposed derivatives were calculated at M06/6-31G(d,p) functional. All the designed compounds have smaller energy bandgaps than the HPTC compound. Moreover, the designed compounds exhibited larger global softness values than the reference. The absorption maxima of HPTC2, HPTC3, and HPTC7 are blue shifted with respect to HPTC. NBO analysis revealed that prolonged hyper conjugative associations and strong interactions between the donor (π) and acceptor (π*) moieties play a crucial part in their stabilization. The FMO and NBO findings supported the NLO responses of entitled compounds, and consequently, the linear and nonlinear properties of designed derivatives elevate compared to the reference molecule. Promisingly, the NLO response for HPTC7 comprises of highest values of <α>, β_total and < γ > as 1.92 × 10^?22 esu, 1.95 × 10^?27 esu, and 4.69 × 10⁷ (a.u). This NLO behavior shows push–pull NLO chromophores for HPTC7 predicting its role in pursuing NLO materials for optoelectronic applications.     

Subset representations and eigenvalues of the universal intertwining matrix

We solve a combinatorial question concerning eigenvalues of the universal intertwining endomorphism of a subset representation.     

Stability of Multiplier Ideal Sheaves

Chinese Annals of Mathematics, Series B - In the present article, the authors find and establish stability of multiplier ideal sheaves, which is more general than strong openness.     

A new idea to construct the fractal interpolation function

The aim of this paper is to present a new idea to construct the nonlinear fractal interpolation function, in which we exploit the Matkowski and the Rakotch fixed point theorems. Our technique is different from the methods presented in the previous literatures.     

Generalized gradient approximation adjusted to transition metals properties: Key roles of exchange and local spin density

Perdew-Burke-Ernzerhof (PBE) and PBE adapted for solids (PBEsol) are exchange-correlation (xc) functionals widely used in density functional theory simulations. Their differences are the exchange, μ, and correlation, β, coefficients, causing PBEsol to lose the Local Spin Density (LSD) response. Here, the μ/β two-dimensional (2D) accuracy landscape is analyzed between PBE and PBEsol xc functional limits for 27 transition metal (TM) bulks, as well as for 81 TM surfaces. Several properties are analyzed, including the shortest interatomic distances, cohesive energies, and bulk moduli for TM bulks, and surface relaxation degree, surface energies, and work functions for TM surfaces. The exploration, comparing the accuracy degree with respect experimental values, reveals that the found xc minimum, called VV, being a PBE variant, represents an improvement of 5% in mean absolute percentage error terms, whereas this improvement reaches ~11% for VVsol, a xc resulting from the restoration of LSD response in PBEsol, and so regarded as its variant.     

Dinuclear gold (I)-di-N-heterocyclic carbene complexes: syntheses,structure, density functional theory calculation and photoluminescence study

The synthesis and characterizations for a series of dinuclear gold (I)-di-NHC complexes, 1–8 through the trans-metalation method of their respective silver (I)-di-NHC complexes, i–viii are reported (where NHC = N-heterocyclic carbene). The successful complexation of a series of unusual non-symmetrical and symmetrical di-NHC ligands, 3,3'-(ethane-1,2-diyl)-1-alkylbenzimidazolium-1'-butylbenzimidazolium (with alkyl = methyl, ethyl, propyl, butyl, pentyl, hexyl, heptyl, benzyl) with the gold (I) ions are suggested by elemental analysis, Fourier transform-infrared, ¹H- and ¹³C-NMR data. The ¹³C-NMR spectra of 1–8 show a singlet sharp peak in the range of 190.00–192.00 ppm, indicating the presence of a carbene carbon that bonded to the gold (I) ion. From single crystal X-ray diffraction data, the structure of complex 6 with the formula of [di-NHC-Au (I)]₂·2PF₆ is obtained [where NHC = 3,3'-(ethane-1,2-diyl)-1-hexylbenzimidazolium-1'-butylbenzimidazolium]. The photophysical study in solid state of 6 displays an intense photoluminescence with a strong emission maxima, λ_em = 480 nm, upon excitation at 340 nm at room temperature. Interestingly, the emission maximum at 77 K shows a structural character with a strong peak at 410 nm, a medium at 433 nm and a weak at 387 nm, accompanied by a tail band to about 500 nm.     

Molecular structure,vibrational spectroscopic and chemical reactivity of nematogenic p-n-alkylbenzoic acids: A comparative computational analysis

Molecular structure and vibrational spectroscopic studies of higher homologous series nematogenic p-n-alkylbenzoic acids (nBAC) that have 6 (6BAC) and 7 (7BAC) carbon atoms in the alkyl chain have been investigated using the Density Functional Becke3-Lee-Yang-Parr (B3LYP) level with the basis set 6-31++G (d.p) and Hartree Fock (HF) with the same basis set. The observed vibrational spectra has been resolved and assigned in detail for comparision with both the molecules. These results indicate that DFT and HF values are slightly different at both the levels. A comparision of chemical reactivity such as HOMO (E_H), LUMO (E_L) energies, energy gap (E_g), ionization energy (I), electron affinity (A), electro negativity (χ), chemical hardness (η), electronic chemical potential (μ), electrophilicity index (ω), and softness (S) has been made. It has been observed that the decrement has occurred in the energy band gap value of isolated molecule with increment in alkyl chain length. This provides valuable information regarding enhancing the stability of liquid crystal materials by maintaining the conductivity.