Fractal and multifractal properties of a family of fractal networks

In this work, we study the fractal and multifractal properties of a family of fractal networks introduced by Gallos {\it et al.} ({\it Proc. Natl. Acad. Sci. U.S.A.}, 2007, {\bf 104}: 7746). In this fractal network model, there is a parameter $e$ which is between $0$ and $1$, and allows for tuning the level of fractality in the network. Here we examine the multifractal behavior of these networks, dependence relationship of fractal dimension and the multifractal parameters on the parameter $e$. First, we find that the empirical fractal dimensions of these networks obtained by our program coincide with the theoretical formula given by Song {\it et al.} ( {\it Nat. Phys}, 2006, {\bf 2}: 275). Then from the shape of the $\tau(q)$ and $D(q)$ curves, we find the existence of multifractality in these networks. Last, we find that there exists a linear relationship between the average information dimension $$ and the parameter $e$.Comment: 12 pages, 7 figures, accepted by J. Stat. Mec

Multifractal analysis of weighted networks by a modified sandbox algorithm

Complex networks have attracted growing attention in many fields. As a generalization of fractal analysis, multifractal analysis (MFA) is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. Some algorithms for MFA of unweighted complex networks have been proposed in the past a few years, including the sandbox (SB) algorithm recently employed by our group. In this paper, a modified SB algorithm (we call it SBw algorithm) is proposed for MFA of weighted networks.First, we use the SBw algorithm to study the multifractal property of two families of weighted fractal networks (WFNs): "Sierpinski" WFNs and "Cantor dust" WFNs. We also discuss how the fractal dimension and generalized fractal dimensions change with the edge-weights of the WFN. From the comparison between the theoretical and numerical fractal dimensions of these networks, we can find that the proposed SBw algorithm is efficient and feasible for MFA of weighted networks. Then, we apply the SBw algorithm to study multifractal properties of some real weighted networks ---collaboration networks. It is found that the multifractality exists in these weighted networks, and is affected by their edge-weights.Comment: 15 pages, 6 figures. Accepted for publication by Scientific Report

Solving and Completing the Rabi-Stark Model in the Ultrastrong Coupling Regime

In this work,we employ a unitary transformation with a suitable parameter to convert the quantum Rabi-Stark model into a Jaynes-Cummings-like model. Subsequently, we derive the analytical energy spectra in the ultrastrong coupling regime. The energy spectra exhibit a phenomenon known as spectral collapse, indicating the instability of the model due to the unboundedness of its energy from below at higher coupling parameters. To stabilize the Rabi-Stark model, we introduce a nonlinear photon-photon interaction term. We then compare the modified model with the original model in the classical oscillator (CO) limit. Interestingly, we observe a regular "staircase" pattern in the mean photon number of the ground state. This pattern exhibits a fixed slope and equal step width, which we determine analytically. Moreover, we analytically determine the phase boundary, which slightly differs from that in the original Rabi-Stark model. These findings offer insights into the investigation of those superradiant phase transitions that are unbounded from below due to the phenomenon of spectral collapse.Comment: 11 pages, 7 figure

2-(Dibutyl­amino)-3-(4-fluoro­phen­yl)-5,6,7,8-tetra­hydro-7-methyl-6,8-di­phenyl­pyridine­[3′,4′:2,3]thieno[5,4-d]pyrimidin-4(3H)-one

In the crystal structure of the title compound, C36H39FN4OS, the two fused rings of the thienopyrimidine system are coplanar. The 4-fluoro­phenyl ring is twisted with respect to the heterocyclic pyrimidinone ring by 67.21 (14)°. The piperidine ring shows a half-chair conformation. One of the n-butyl chains is disordered equally over two sites. The crystal packing is stabilized by C—H⋯O hydrogen bonds

Non-Gaussian features from the inverse volume corrections in loop quantum cosmology

In this paper we study the non-Gaussian features of the primordial fluctuations in loop quantum cosmology with the inverse volume corrections. The detailed analysis is performed in the single field slow-roll inflationary models. However, our results reflect the universal characteristics of bispectrum in loop quantum cosmology. The main corrections to the scalar bispectrum come from two aspects: one is the modifications to the standard Bunch-Davies vacuum, the other is the corrections to the background dependent variables, such as slow-roll parameters. Our calculations show that the loop quantum corrections make $f_{{\rm NL}}$ of the inflationary models increase 0.1%. Moreover, we find that two new shapes arise, namely $\mathcal F_{1}$ and $\mathcal F_{2}$. The former gives a unique loop quantum feature which is less correlated with the local, equilateral and single types, while the latter is highly correlated with the local one.Comment: matched to the published version. 30 pages, 4 figure

Production of proton-rich nuclei around Z=84-90 in fusion-evaporation reactions

Within the framework of the dinuclear system model, production cross sections of proton-rich nuclei with charged numbers of Z=84-90 are investigated systematically. Possible combinations with the $^{28}$Si, $^{32}$S, $^{40}$Ar bombarding the target nuclides $^{165}$Ho, $^{169}$Tm, $^{170-174}$Yb, $^{175,176}$Lu, $^{174,176-180}$Hf and $^{181}$Ta are analyzed thoroughly. The optimal excitation energies and evaporation channels are proposed to produce the proton-rich nuclei. The systems are feasible to be constructed in experiments. It is found that the neutron shell closure of N=126 is of importance during the evaporation of neutrons. The experimental excitation functions in the $^{40}$Ar induced reactions can be nicely reproduced. The charged particle evaporation is comparable with neutrons in cooling the excited proton-rich nuclei, in particular for the channels with $\alpha$ and proton evaporation. The production cross section increases with the mass asymmetry of colliding systems because of the decrease of the inner fusion barrier. The channels with pure neutron evaporation depend on the isotopic targets. But it is different for the channels with charged particles and more sensitive to the odd-even effect.Comment: 15 pages, 10 figures. arXiv admin note: text overlap with arXiv:0803.1117, arXiv:0707.258