On concentration of solution to a Schrödinger logarithmic equation with deepening potential well

In this work, we prove the existence of positive solution for the following class of problems where λ>0 and is a potential satisfying some conditions. Using the variational method developed by Szulkin for functionals, which are the sum of a C¹ functional with a convex lower semicontinuous functional, we prove that for each large enough λ>0, there exists a positive solution for the problem, and that, as λ→+∞, such solutions converge to a positive solution of the limit problem defined on the domain Ω=int(V^?1({0})).     

High iso Aldehyde Selectivity in the Hydroformylation of Short‐Chain Alkenes

The hydroformylation of propene to give predominantly iso‐butanal has been achieved; class‐leading selectivity is possible even at higher temperatures that deliver fast conversion. Racemic rhodium complexes of bidentate phospholane phosphites derived from tropos‐biphenols and unusual solvent systems are the key to the selectivity observed.     

Existence of nontrivial solutions for fractional Schrödinger equations with critical or supercritical growth

In this paper, we study the following fractional Schrödinger equation with critical or supercritical growth where 0 < s < 1, N > 2s, λ > 0, , , ( ? Δ)^s denotes the fractional Laplacian of order s and f is a continuous superlinear but subcritical function. Under some suitable conditions, we prove that the equation has a nontrivial solution for small λ > 0 by variational methods. Our main contribution is related to the fact that we are able to deal with the case .     

Lower bound for the ratios of eigenvalues of Schrödinger with nonpositive single‐barrier potentials

Horváth and Kiss (Proc. Amer. Math. Soc., 2005) proved the upper bound estimate for Dirichlet eigenvalue ratios of the Schrödinger problem ?y^′′ + q(x)y = λy with nonnegative and single‐well potential q. In this paper, we prove that if q(x) is a nonpositive, continuous, and single‐barrier potential, then for λ_n > λ_m≥ ? 2q^?, where . In particular, if q(x) satisfies the additional condition , then λ₁ > 0 and for n > m ≥ 1. For this result, we develop a new approach to study the monotonicity of the modified Prüfer angle function.     

Weighted estimates for the multilinear maximal function on the upper half‐spaces

For a general dyadic grid, we give a Calderón–Zygmund type decomposition, which is the principle fact about the multilinear maximal function on the upper half‐spaces. Using the decomposition, we study the boundedness of . We obtain a natural extension to the multilinear setting of Muckenhoupt's weak‐type characterization. We also partially obtain characterizations of Muckenhoupt's strong‐type inequalities with one weight. Assuming the reverse Hölder's condition, we get a multilinear analogue of Sawyer's two weight theorem. Moreover, we also get Hytönen–Pérez type weighted estimates.     

Ce掺杂6H-SiC电子结构和光学性质的第一性原理研究

采用基于密度泛函理论的第一性原理计算方法,对未掺杂及Ce掺杂6H-SiC的电子结构和光学性质进行理论计算.计算结果表明,未掺杂6H-SiC是间接带隙半导体,其禁带宽度为2.045 eV,掺杂Ce元素,带隙宽度下降为0.812 eV.未掺杂6H-SiC在价带的低能区,Si-3s、C-2s电子轨道对态密度的贡献较大,在价带的高能区,主要是由Si-3p、Si-3s、C-2p态组成.掺杂后Ce原子的4f轨道主要贡献在导带部分,掺杂后电导率提高.未掺杂时,只有一个介电峰,是价带电子跃迁到导带电子所致,掺杂后有两个介电峰,第一个介电峰是由于导带电子跃迁到Ce原子4f轨道上产生,第二个峰是价带电子向导带电子跃迁产生.未掺杂6H-SiC,在能量为10.31 eV处吸收系数达到最大值,掺杂后在能量为6.57 eV处,吸收系数达到最大值.     

Photoluminescence of green nanophosphors Sr2MgSi2O7 doped with Tb3+ under 374-nm excitation

A series of Sr₂MgSi₂O₇:Tb³⁺ nanophosphors is prepared using a high-temperature solid-state reaction. The x-ray diffraction patterns show that the crystal structure of the sample is not significantly affected by Tb³⁺ ions. However, the images of the scanning electron microscope illustrate that the average size of nanoparticles becomes larger with the increase of Tb³⁺ concentration. Unlike earlier investigations on down-conversion emission of Tb³⁺ ion excited by deep ultraviolet light, in this work, the photoluminescence characteristics of Sr₂MgSi₂O₇ nanophosphors doped with different Tb³⁺ concentrations are analyzed under 374-nm excitations. The intense green emission at 545 nm is observed at an optimal doping concentration of 1.6 mol%. The main reason for the concentration quenching is due to the electric dipole-electric dipole interaction among Tb³⁺ ions.     

A note on the finite fractal dimension of the global attractors for dissipative nonlinear Schrödinger‐type equations

In this article, we present, throughout two basic models of damped nonlinear Schrödinger (NLS)–type equations, a new idea to bound from above the fractal dimension of the global attractors for NLS‐type equations. This could answer the following open issue: consider, for instance, the classical one‐dimensional cubic nonlinear Schrödinger equation $$u_t+i{u}_{xx}+i|u{|}^{2}u+\gamma u=f,f\in {\mathrm{??}}^2(?).$$ “How can we bound the fractal dimension of the associate global attractor without the need to assume that the external forcing term f has some decay at infinity (that is belonging to some weighted Lebesgue space)?”     

Multivariate optimization and comparison between conventional extraction (CE) and ultrasonic-assisted extraction (UAE) of carotenoid extraction from cashew apple

Carotenoids are an essential component of cashew and can be used in pharmaceuticals, cosmetics, natural pigment, food additives, among other applications. The present work focuses on optimizing and comparing conventional and ultrasound-assisted extraction methods. Every optimization step took place with a 1:1 (w:w) mixture of yellow and red cashew apples lyophilized and ground in a cryogenic mill. A Simplex-centroid design was applied for both methods, and the solvents acetone, methanol, ethanol, and petroleum ether were evaluated. After choosing the extractor solvent, a central composite design was applied to optimize the sample mass (59–201 mg) and extraction time (6–34 min). The optimum conditions for the extractor solvent were 38% acetone, 30% ethanol, and 32% petroleum ether for CE and a mixture of 44% acetone and 56% methanol for UAE. The best experimental conditions for UAE were a sonication time of 19 min and a sample mass of 153 mg, while the CE was 23 min and 136 mg. Comparing red and yellow cashews, red cashews showed a higher carotenoid content in both methodologies. The UAE methodology was ca. 21% faster, presented a more straightforward composition of extracting solution, showed an average yield of superior carotenoid content in all samples compared to CE. Therefore, UAE has demonstrated a simple, efficient, fast, low-cost adjustment methodology and a reliable alternative for other applications involving these bioactive compounds in the studied or similar matrix.     

Site Occupancies,Electron–Vibration Interaction and Energy Transfer of CaMgSi2O6: Eu2+, Mn2+ Phosphors for Potential Temperature-Sensing and Anti-counterfeiting Applications

Eu²⁺-, Mn²⁺- and Eu²⁺−Mn²⁺-doped CaMgSi₂O₆ phosphors have been prepared by a high-temperature solid-state reaction. Systematic investigation of the concentration- and temperature-dependent luminescence of Mn²⁺ showed that Mn²⁺ ions occupy two distinct sites in CaMgSi₂O₆. Electron–vibration interaction (EVI) analyses of Mn²⁺ ions revealed Huang–Rhys factors of 4.73 and 2.82 as well as effective phonon energies of 313 and 383 cm⁻¹ for the two sites. Eu²⁺−Mn²⁺ energy transfer is also discussed, and its efficiency is estimated by lifetime and luminescence spectra. The different thermal quenching behaviours of Eu²⁺ and Mn²⁺, the distinct emission colours of Eu²⁺ (blue, band peak at ∼451 nm) and Mn²⁺ (yellow–red range, band peaks at ∼583 and 693 nm) endow the co-doped samples with potential applications in luminescence thermometry and temperature-/excitation wavelength-responsive dual anti-counterfeiting.