Normalized Sombor Indices as Complexity Measures of Random Networks.

We perform a detailed computational study of the recently introduced Sombor indices on random networks. Specifically, we apply Sombor indices on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the network, scale with the average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random networks and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the adjacency matrix.

Information-Length Scaling in a Generalized One-Dimensional Lloyd's Model.

We perform a detailed numerical study of the localization properties of the eigenfunctions of one-dimensional (1D) tight-binding wires with on-site disorder characterized by long-tailed distributions: For large ϵ , P ( ϵ ) ∼ 1 / ϵ 1 + α with α ∈ ( 0 , 2 ] ; where ϵ are the on-site random energies. Our model serves as a generalization of 1D Lloyd's model, which corresponds to α = 1 . In particular, we demonstrate that the information length ß of the eigenfunctions follows the scaling law ß = γ x / ( 1 + γ x ) , with x = ξ / L and γ ≡ γ ( α ) . Here, ξ is the eigenfunction localization length (that we extract from the scaling of Landauer's conductance) and L is the wire length. We also report that for α = 2 the properties of the 1D Anderson model are effectively reproduced.

Scaling properties of the action in the Riemann-Liouville fractional standard map.

The Riemann-Liouville fractional standard map (RL-fSM) is a two-dimensional nonlinear map with memory given in action-angle variables (I,θ). The RL-fSM is parameterized by K and α∈(1,2], which control the strength of nonlinearity and the fractional order of the Riemann-Liouville derivative, respectively. In this work we present a scaling study of the average squared action 〈I^{2}〉 of the RL-fSM along strongly chaotic orbits, i.e., for K≫1. We observe two scenarios depending on the initial action I_{0}, I_{0}≪K or I_{0}≫K. However, we can show that 〈I^{2}〉/I_{0}^{2} is a universal function of the scaled discrete time nK^{2}/I_{0}^{2} (n being the nth iteration of the RL-fSM). In addition, we note that 〈I^{2}〉 is independent of α for K≫1. Analytical estimations support our numerical results.

Topological versus spectral properties of random geometric graphs.

In this work we perform a detailed statistical analysis of topological and spectral properties of random geometric graphs (RGGs), a graph model used to study the structure and dynamics of complex systems embedded in a two-dimensional space. RGGs, G(n,ℓ), consist of n vertices uniformly and independently distributed on the unit square, where two vertices are connected by an edge if their Euclidian distance is less than or equal to the connection radius ℓ∈[0,sqrt[2]]. To evaluate the topological properties of RGGs we chose two well-known topological indices, the Randic index R(G) and the harmonic index H(G). We characterize the spectral and eigenvector properties of the corresponding randomly weighted adjacency matrices by the use of random matrix theory measures: the ratio between consecutive eigenvalue spacings, the inverse participation ratios, and the information or Shannon entropies S(G). First, we review the scaling properties of the averaged measures, topological and spectral, on RGGs. Then we show that (i) the averaged-scaled indices, 〈R(G)〉 and 〈H(G)〉, are highly correlated with the average number of nonisolated vertices 〈V_{×}(G)〉; and (ii) surprisingly, the averaged-scaled Shannon entropy 〈S(G)〉 is also highly correlated with 〈V_{×}(G)〉. Therefore, we suggest that very reliable predictions of eigenvector properties of RGGs could be made by computing topological indices.

Phase transition of two-dimensional chiral supramolecular nanostructure tuned by electrochemical potential.

Molecular chiralty and phase transition of p-phenylenedi(alpha-cyanoacrylicacid) di-n-ethyl ester (p-CPAEt) assembled on Au(111) have been studied in the electric double layer region in 0.1 M HClO(4) by electrochemical scanning tunneling microscopy (ECSTM) technique. Three types of chiral supramolecular nanostructures were resolved at differently charged interfaces. Within a potential range (0.65 V < E < 0.8 V, region I), a close-packed physisorbed adlayer of chiral stripe pattern, with the (3 x 6) structure, has been observed. At more negative potential (0.2 V < E < or = 0.65 V, region II), the stripe patterns gradually dissolved, and two types of new chiral network structures (3 square root(7) x 4 square root(7)) and (3 square root(7) x 3 square root(7)) evolved on reconstructed and unreconstructed surfaces, respectively. On the basis of the high-resolution STM images, it was tentatively proposed that three types of chiral supramolecular nanostructures were formed by two-dimensional adsorption-induced chiral p-CPAEt species together with lateral hydrogen-bonding interaction (C-H...N[triple bond]C). Intriguingly, ECSTM images allow in situ monitoring of the phase transition process of these chiral adlayers driven by the electrochemical potential. The detailed dynamic results showed that the chiral two-dimensional adlayers could be reversibly tuned purely by the applied electrode potential.

Chromatographic and electrochemical determination of quercetin and kaempferol in phytopharmaceuticals.

An NP-HPLC method both with diode-array (DAD) and electrochemical detection (ED) was developed and validated for the determination of quercetin and kaempferol, the principal active constituents in phytopharmaceuticals of Ginkgo Biloba. Calculated retention of the two flavonoids was contrasted with experimental values in five different reversed phase columns for methanol-water, acetonitrile-water, THF-water and dioxane-hexane binary mixtures as mobile phases. The capacity factor k, selectivity alpha and asymmetry factor F were evaluated and compared in DAD-RP-HPLC, DAD-NP-HPLC, ED-RP-HPLC and ED-NP-HPLC. The methods were used for the quantitative analysis of acid hydrolyzed extracts of tablet phytopharmaceuticals. Calibration curves were linear within the range 10 and 40 microg ml(-1) for the DAD and 10-270 microg ml(-1) for the ED, whereby limits of detection ranged from 0.5 microg ml(-1) (quercetin) to 0.1 microg ml(-1) (kaempferol). The electrochemical method based on differential pulse voltammetry (DPV) with a C-PVC electrode resolved the quercetin and kaempferol peaks and exhibited a two orders higher sensitivity in comparison with a carbon fiber electrode. DPV calibration curves were linear within the range 96-300 microg ml(-1) for quercetin and 68-960 microg ml(-1) for kaempferol. The respective oxidation peaks appeared at 462 and 518+/-2 mV and were used in the direct determination of quercetin in extracts of commercial phytopharmaceuticals.

Scaling properties of discontinuous maps.

We study the scaling properties of discontinuous maps by analyzing the average value of the squared action variable I(2). We focus our study on two dynamical regimes separated by the critical value K(c) of the control parameter K: the slow diffusion (K < K(c)) and the quasilinear diffusion (K > K(c)) regimes. We found that the scaling of I(2) for discontinuous maps when K ≪ K(c) and K ≫ K(c) obeys the same scaling laws, in the appropriate limits, as Chirikov's standard map in the regimes of weak and strong nonlinearity, respectively. However, due to the absence of Kolmogorov-Arnold-Moser tori, we observed in both regimes that I(2) ∝ nK(ß) for n ≫ 1 (n being the nth iteration of the map) with ß ≈ 5/2 when K ≪ K(c) and ß ≈ 2 for K ≫ K(c).