Search

In this article, we study the fractional critical Choquard equation with a nonlocal perturbation: (−Δ)su=λu+α(Iμ*∣u∣q)∣u∣q−2u+(Iμ*∣u∣2μ,s*)∣u∣2μ,s*−2u,inRN,{\left(-{\Delta })}^{s}u=\lambda u+\alpha \left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{q}){| u| }^{q-2}u+\left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{{2}_{\mu ,s}^{* }}){| u| }^{{2}_{\mu ,s}^{* }-2}u,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, having prescribed mass ∫RNu2dx=c2,\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{2}{\rm{d}}x={c}^{2}, where s∈(0,1),N>2s,00,c>0s\in \left(0,1),N\gt 2s,0\lt \mu \lt N,\alpha \gt 0,c\gt 0, and Iμ(x){I}_{\mu }\left(x) is the Riesz potential given by Iμ(x)=Aμ∣x∣μwithAμ=Γμ22N−μπN⁄2ΓN−μ2,{I}_{\mu }\left(x)=\frac{{A}_{\mu }}{{| x| }^{\mu }}\hspace{1em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{A}_{\mu }=\frac{\Gamma \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{\mu }{2}\right)}{{2}^{N-\mu }{\pi }^{N/2}\Gamma \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{N-\mu }{2}\right)}, and 2N−μN

In this paper, we study the fractional Schrödinger-Poisson system

Systematically orchestrating fundamental building blocks into intricate high‐dimensional molecular assemblies at molecular level is imperative for multifunctionality integration. However, this remains a formidable task in crystal engineering due to the dynamic nature of inorganic building blocks. Herein, we develop a multi‐template‐guided strategy to control building blocks. The coordination modes of ligands and the spatial hindrance of anionic templates are pivotal in dictating the overall structures. Flexible multi‐dentate linkers selectively promote the formation of oligomeric assembly ([TeO3(Mo2O2S2)3O2(OH)(C5O2H7)3]4− {TeMo6}) into tetrahedral cages ([(TeO3)4(Mo2O2S2)12(OH)12(C9H9O4P)6]8− {Te4Mo24} and [(AsO4)4(Mo2O2S2)12(OH)12(C9H9O6)4]12− {As4Mo24}), while steric hindrance from anionic templates further assists in assembling cages into an open quadruply twisted Möbius nanobelt ([(C6H5O3P)8(Mo2O2S2)24(OH)24(C8H10O4)12]16− {P8Mo48}). Among these structures, the hydrophilic‐hydrophobic hybrid cage {Te4Mo24} emerges as an exemplary molecular model for proton conduction and serves as a prototype for humidity gradient‐based power generators (HGPGs). The Te4Mo24‐PVDF‐based HGPG (PVDF=Poly(vinylidene fluoride)) exhibits notable stability and power generation, yielding an open‐circuit voltage of 0.51 V and a current density of 77.8 nA cm−2 at room temperature and 90 % relative humidity (RH). Further insights into the interactions between water molecules and microscale molecules within the generator are achieved through molecular dynamics simulations. This endeavor unveils a universal strategy for synthesizing multifunctional integration molecules. [ABSTRACT FROM AUTHOR]

We present a scheme based on artificial neural networks (ANNs) to estimate the line-of-sight velocities of individual galaxies from an observed redshift–space galaxy distribution. We find an estimate of the peculiar velocity at a galaxy based on galaxy counts and barycentres in shells around it. By training the network with environmental characteristics, such as the total mass and mass centre within each shell surrounding every galaxy in redshift space, our ANN model can accurately predict the line-of-sight velocity of each individual galaxy. When this velocity is used to eliminate the RSD effect, the two-point correlation function (TPCF) in real space can be recovered with an accuracy better than 1 per cent at s > 8 |$\, h^{-1}\, \mathrm{Mpc}$|⁠ , and 4 per cent on all scales compared to ground truth. The real-space power spectrum can be recovered within 3 per cent on k < 0.5 |$\, \mathrm{Mpc}^{-1}\, h$|⁠ , and less than 5 per cent for all k modes. The quadrupole moment of the TPCF or power spectrum is almost zero down to s = 10 |$\, h^{-1}\, \mathrm{Mpc}$| or all k modes, indicating an effective correction of the spatial anisotropy caused by the RSD effect. We demonstrate that on large scales, without additional training with new data, our network is adaptable to different galaxy formation models, different cosmological models, and mock galaxy samples at high-redshifts and high biases, achieving less than 10 per cent error for scales greater than 15 |$\, h^{-1}\, \mathrm{Mpc}$|⁠. As it is sensitive to large-scale densities, it does not manage to remove Fingers of God in large clusters, but works remarkably well at recovering real-space galaxy positions elsewhere. Our scheme provides a novel way to predict the peculiar velocity of individual galaxies, to eliminate the RSD effect directly in future large galaxy surveys, and to reconstruct the three-dimensional cosmic velocity field accurately. [ABSTRACT FROM AUTHOR]

In this paper we study the fractional Schrödinger–Poisson system ε 2 s (− Δ) s u + V (x) u = ϕ | u | 2 s * − 3 u + | u | 2 s * − 2 u , ε 2 s (− Δ) s ϕ = | u | 2 s * − 1 , x ∈ R 3 , where s ∈ (0, 1), ɛ > 0 is a small parameter, 2 s * = 6 3 − 2 s is the critical Sobolev exponent and V ∈ L 3 2 s ( R 3 ) is a nonnegative function which may be zero in some regions of R 3 , e.g., it is of the critical frequency case. By virtue of a new global compactness lemma, and the Lusternik–Schnirelmann category theory, we relate the number of bound state solutions with the topology of the zero set where V attains its minimum for small values of ɛ. [ABSTRACT FROM AUTHOR]

A series of pyrimidine derivatives were constructed by replacing hydrogen atoms with nitro groups. To explore their thermal stability and kinetic stability, heat of formation (HOF) and dissociation energies (BDE) are calculated based on the density functional theory with the B3PW91/6-311+G(d,p)//MP2/6-311+G(d,p) method. Based on our calculations, all derivatives are confirmed stable enough to be synthesized in laboratory. To explore the detonation characters, the detonation velocity and detonation pressure are calculated by using the empirical Kamlet-Jacobs equation and good detonation parameters were confirmed for most of title molecules. Based on our calculations, four molecules (2,4,5,6-tetranitropyrimidine, 4,5,6-trinitropyrimidine, 2,4,6-trinitropyrimidine, 2,4,5-trinitropyrimidine) are screened out as potential high-energy-density materials for further research. [ABSTRACT FROM AUTHOR]