Phase separation in polydisperse rod–coil block copolymers

The stability limits of the homogeneous state of melts of rod–coil RC$RC$, R_C2$R{C}_2$, and CRC$CRC$ polydisperse block copolymers have been investigated in the framework of the weak segregation theory. It was assumed that the number of units in either the rod-like R$R$ or the flexible C$C$ block is a random variable distributed by the Schulz–Zimm distribution. Inspection of the spinodal curves shows that the copolymer melts with polydisperse rigid blocks are less stable with respect to formation of the nematic state than melts with the monodisperse ones. If flexible C$C$ blocks are polydisperse the homogeneous state of a rod–coil melt is less stable against microphase separation than the homogeneous state of monodisperse melt of the same architecture.     

Density outbursts in a food web model with a closed nutrient cycle

A spatial three level food web model with a closed nutrient cycle is presented and analyzed via Monte Carlo simulations. The food web consists of three trophic levels. The basal level species (called resources  , R$R$) corresponds to primary producers in real ecosystems. The species at an intermediate level (consumers  , C$C$) relates to herbivores. It feeds on the resources. The consumers themselves constitute food for the top level species (predators  , P$P$), which corresponds to carnivores. The remains of the consumers and predators (detritus  , D$D$) provide nutrient for the resources. The time evolution of the model reveals two asymptotic states: an absorbing one with all species being extinct, and a coexisting one, in which concentrations of all species are non-zero. There are two possible ways for the system to reach the absorbing state. In some cases the densities increase very quickly at the beginning of a simulation and then decline slowly and almost monotonically. In others, well pronounced peaks in the R$R$, C$C$ and D$D$ densities appear regularly before the extinction. Those peaks correspond to density outbursts (waves) traveling through the system. We investigate the mechanisms leading to the waves. In particular, we show that the percolation of the detritus (i.e. the accumulation of nutrients) is necessary for the emergence of the waves. Moreover, our results corroborate the hypothesis that top-level predators play an essential role in maintaining the stability of a food web (top-down control).     

Intertwining operators for Sklyanin algebra and elliptic hypergeometric series

Intertwining operators for infinite-dimensional representations of the Sklyanin algebra with spins ?$?$ and −?−1$-?-1$ are constructed using the technique of intertwining vectors for elliptic L$L$-operator. They are expressed in terms of elliptic hypergeometric series with operator argument. The intertwining operators obtained (W$W$-operators) serve as building blocks for the elliptic R$R$-matrix which intertwines tensor product of two L$L$-operators taken in infinite-dimensional representations of the Sklyanin algebra with arbitrary spin. The Yang–Baxter equation for this R$R$-matrix follows from simpler equations of the star–triangle type for the W$W$-operators. A natural graphic representation of the objects and equations involved in the construction is used.     

On the geometry of prequantization spaces

Given a Poisson (or more generally Dirac) manifold P$P$, there are two approaches to its geometric quantization: one involves a circle bundle Q$Q$ over P$P$ endowed with a Jacobi (or Jacobi–Dirac) structure; the other one involves a circle bundle with a (pre)contact groupoid structure over the (pre)symplectic groupoid of P$P$. We study the relation between these two prequantization spaces. We show that the circle bundle over the (pre)symplectic groupoid of P$P$ is obtained from the Lie groupoid of Q$Q$ via an S¹$S^1$ reduction that preserves both the Lie groupoid and the geometric structures.     

Can density functional theory make use of experimentally determined ground-state electron densities via the Dirac density matrix for molecules of biological interest?

More than four decades ago, March and Murray gave a perturbation theory of the single-particle(s) Dirac density matrix γs(r,r^′)${\gamma }_s(r,r^{\prime })$ to all orders in a given one-body potential energy V(r)$V(r)$. However, for density functional theory in orbital-free form, one requires the functional γs[ρ]${\gamma }_s[\rho ]$ where ρ(r)$\rho (r)$ is the ground-state electron density. Therefore, in the present study, a first-order non-linear differential equation is proposed for γs${\gamma }_s$ in terms of ρ(r)$\rho (r)$ and ∇ρ(r)$\mathrm{\nabla }\rho (r)$, plus the single-particle kinetic energy. Since this latter quantity is itself known to be a functional of ρ  , the existence of such an equation for γs${\gamma }_s$ would be a significant step along the road to determining the desired functional γs[ρ]${\gamma }_s[\rho ]$. As yet, we have succeeded in giving a rigorous proof of the proposed differential equation for γs(r,r^′)${\gamma }_s(r,r^{\prime })$ only for one- and two-level molecules. If it is subsequently proved for an arbitrary number of levels, which we believe should be possible, it would then allow γs${\gamma }_s$ to be calculated for molecules of biological interest, from experimentally measured ground-state densities ρ(r)$\rho (r)$, as the approach is entirely orbital-free.     

Dust ion-acoustic rogue waves in a three-species ultracold quantum dusty plasmas

Dust ion-acoustic (DIA) rogue waves are reported for a three-component ultracold quantum dusty plasma comprised of inertialess electrons, inertial ions, and negatively charged immobile dust particles. The nonlinear Schrödinger (NLS) equation appears for the low frequency limit. Modulation instability (MI) of the DIA waves is analyzed. Influence of the modulation wave number, ion-to-electron Fermi temperature ratio ρ$\rho$ and dust-to-ion background density ratio N_d$N_d$ on the MI growth rate is discussed. The first- and second-order DIA rogue-wave solutions of the NLS equation are examined numerically. It is found that the enhancement of N_d$N_d$ and carrier wave number can increase the envelope rogue-wave amplitudes. However, the increase of ρ$\rho$ reduces the envelope rogue-wave amplitudes.     

Weak mirror symmetry of complex symplectic Lie algebras

A complex symplectic structure on a Lie algebra h$\mathfrak{h}$ is an integrable complex structure J$J$ with a closed non-degenerate (2,0)$(2,0)$-form. It is determined by J$J$ and the real part Ω$\Omega$ of the (2,0)$(2,0)$-form. Suppose that h$\mathfrak{h}$ is a semi-direct product g?V$\mathfrak{g}?V$, and both g$\mathfrak{g}$ and V$V$ are Lagrangian with respect to Ω$\Omega$ and totally real with respect to J$J$. This note shows that g?V$\mathfrak{g}?V$ is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of Ω$\Omega$ and J$J$ are isomorphic.     

Two charges on a plane in a magnetic field: II. Moving neutral quantum system across a magnetic field

The moving neutral system of two Coulomb charges on a plane subject to a constant magnetic field B$B$ perpendicular to the plane is considered. It is shown that the composite system of finite total mass is bound for any center-of-mass momentum P$P$ and magnetic field strength; the energy of the ground state is calculated accurately using a variational approach. Its accuracy is cross-checked in a Lagrange-mesh method for B=1$B=1$  a.u. and in a perturbation theory at small B$B$ and P$P$. The constructed trial function has the property of being a uniform approximation of the exact eigenfunction. For a Hydrogen atom and a Positronium a double perturbation theory in B$B$ and P$P$ is developed and the first corrections are found algebraically. A phenomenon of a sharp change of energy behavior for a certain center-of-mass momentum and a fixed magnetic field is indicated.     

Exact solution of the alternating XXZ spin chain with generic non-diagonal boundaries

The integrable XXZ alternating spin chain with generic non-diagonal boundary terms specified by the most general non-diagonal K$K$-matrices is studied via the off-diagonal Bethe Ansatz method. Based on the intrinsic properties of the fused R$R$-matrices and K$K$-matrices, we obtain certain closed operator identities and conditions, which allow us to construct an inhomogeneous T−Q$T-Q$ relation and the associated Bethe Ansatz equations accounting for the eigenvalues of the transfer matrix.     

An efficient community detection method based on rank centrality

Community detection is a very important problem in social network analysis. Classical clustering approach, K$K$-means, has been shown to be very efficient to detect communities in networks. However, K$K$-means is quite sensitive to the initial centroids or seeds, especially when it is used to detect communities. To solve this problem, in this study, we propose an efficient algorithm K$K$-rank, which selects the top-K$K$ nodes with the highest rank centrality as the initial seeds, and updates these seeds by using an iterative technique like K$K$-means. Then we extend K$K$-rank to partition directed, weighted networks, and to detect overlapping communities. The empirical study on synthetic and real networks show that K$K$-rank is robust and better than the state-of-the-art algorithms including K$K$-means, BGLL, LPA, infomap and OSLOM.     

Dworkin’s argument revisited: Point processes,dynamics, diffraction,and correlations

The setting is an ergodic dynamical system (X,μ)$(X,\mu )$ whose points are themselves uniformly discrete point sets Λ$\Lambda$ in some space R^d${\mathbb{R}}^d$ and whose group action is that of translation of these point sets by the vectors of R^d${\mathbb{R}}^d$. Steven Dworkin’s argument relates the diffraction of the typical point sets comprising X$X$ to the dynamical spectrum of X$X$. In this paper we look more deeply at this relationship, particularly in the context of point processes.     

Effects of uniform rotational flow on predator–prey system

Rotational flow is often observed in lotic ecosystems, such as streams and rivers. For example, when an obstacle interrupts water flowing in a stream, energy dissipation and momentum transfer can result in the formation of rotational flow, or a vortex. In this study, I examined how rotational flow affects a predator–prey system by constructing a spatially explicit lattice model consisting of predators, prey, and plants. A predation relationship existed between the species. The species densities in the model were given as S$S$ (for predator), P$P$ (for prey), and G$G$ (for plant). A predator (prey) had a probability of giving birth to an offspring when it ate prey (plant). When a predator or prey was first introduced, or born, its health state was assigned an initial value of 20 that subsequently decreased by one with every time step. The predator (prey) was removed from the system when the health state decreased to less than zero. The degree of flow rotation was characterized by the variable, R$R$. A higher R$R$ indicates a higher tendency that predators and prey move along circular paths. Plants were not affected by the flow because they were assumed to be attached to the streambed. Results showed that R$R$ positively affected both predator and prey survival, while its effect on plants was negligible. Flow rotation facilitated disturbances in individuals’ movements, which consequently strengthens the predator and prey relationship and prevents death from starvation. An increase in S$S$ accelerated the extinction of predators and prey.     

Probing dark energy using its density instead of its equation of state

The variation of dark energy density with redshift, ρX(z)${\rho }_X(z)$, provides a critical clue to the nature of dark energy. Since ρX(z)${\rho }_X(z)$ depends on the dark energy equation of state wX(z)$w_X(z)$ through an integral, ρX(z)${\rho }_X(z)$ can be constrained more tightly than wX(z)$w_X(z)$ given the same observational data. We demonstrate this explicitly using current type Ia supernova (SN Ia) data [the Tonry/Barris sample], together with the Cosmic Microwave Background (CMB) shift parameter from CMB data (WMAP, CBI, and ACBAR), and the large scale structure (LSS) growth factor from 2dF galaxy survey data. We assume a flat universe, and use Markov Chain Monte Carlo (MCMC) technique in our analysis. We find that, while wX(z)$w_X(z)$ extracted from current data is consistent with a cosmological constant at 68% C.L., ρX(z)${\rho }_X(z)$ (which has far smaller uncertainties) is not. Our results clearly show the advantage of using ρX(z)${\rho }_X(z)$, instead of wX(z)$w_X(z)$, to probe the variation of dark energy.     

Detection of community structure in networks based on community coefficients

Determining community structure in networks is fundamental to the analysis of the structural and functional properties of those networks, including social networks, computer networks, and biological networks. Modularity function Q$Q$, which was proposed by Newman and Girvan, was once the most widely used criterion for evaluating the partition of a network into communities. However, modularity Q$Q$ is subject to a serious resolution limit. In this paper, we propose a new function for evaluating the partition of a network into communities. This is called community coefficient C$C$. Using community coefficient C$C$, we can automatically identify the ideal number of communities in the network, without any prior knowledge. We demonstrate that community coefficient C$C$ is superior to the modularity Q$Q$ and does not have a resolution limit. We also compared the two widely used community structure partitioning methods, the hierarchical partitioning algorithm and the normalized cuts (Ncut) spectral partitioning algorithm. We tested these methods on computer-generated networks and real-world networks whose community structures were already known. The Ncut algorithm and community coefficient C$C$ were found to produce better results than hierarchical algorithms. Unlike several other community detection methods, the proposed method effectively partitioned the networks into different community structures and indicated the correct number of communities.     

On the parallel transport of the Ricci curvatures

Geometrical characterizations are given for the tensor R⋅S$R\cdot S$, where S$S$ is the Ricci tensor   of a (semi-)Riemannian manifold (M,g)$(M,g)$ and R$R$ denotes the curvature operator   acting on S$S$ as a derivation, and of the Ricci Tachibana tensor  _∧g⋅S${\wedge }_g\cdot S$, where the natural metrical operator  _∧g${\wedge }_g$ also acts as a derivation on S$S$. As a combination, the Ricci curvatures   associated with directions on M$M$, of which the isotropy determines that M$M$ is Einstein, are extended to the Ricci curvatures of Deszcz   associated with directions and planes on M$M$, and of which the isotropy determines that M$M$ is Ricci pseudo-symmetric in the sense of Deszcz.     

Skeleton of weighted social network

In the literature of social networks, understanding topological structure is an important scientific issue. In this paper, we construct a network from mobile phone call records and use the cumulative number of calls as a measure of the weight of a social tie. We extract skeletons from the weighted social network on the basis of the weights of ties, and we study their properties. We find that strong ties can support the skeleton in the network by studying the percolation characters. We explore the centrality of w$w$-skeletons based on the correlation between some centrality measures and the skeleton index w$w$ of a vertex, and we find that the average centrality of a w$w$-skeleton increases as w$w$ increases. We also study the cumulative degree distribution of the successive w$w$-skeletons and find that as w$w$ increases, the w$w$-skeleton tends to become more self-similar. Furthermore, fractal characteristics appear in higher w$w$-skeletons. We also explore the global information diffusion efficiency of w$w$-skeletons using simulations, from which we can see that the ties in the high w$w$-skeletons play important roles in information diffusion. Identifying such a simple structure of a w$w$-skeleton is a step forward toward understanding and representing the topological structure of weighted social networks.     

A light scalar in low-scale technicolor

In addition to the narrow spin-one resonances ρT${\rho }_T$, ωT${\omega }_T$ and aT$a_T$ occurring in low-scale technicolor, there will be relatively narrow scalars in the mass range 200 to 600–700 GeV. We study the lightest isoscalar state, σT${\sigma }_T$. In several important respects it is like a heavy Higgs boson with a small vev. It may be discoverable with high luminosity at the LHC where it is produced via weak boson fusion and likely has substantial W⁺W⁻$W^{+}{W}^{-}$ and Z⁰Z⁰$Z^0{Z}^0$ decay modes.