We present a memory-based snowdrift game (MBSG) taking place on networks. We found that, when a lattice is taken to be the underlying structure, the transition of spatial patterns at some critical values of the payoff parameter is observable for both four- and eight-neighbor lattices. The transition points as well as the styles of spatial patterns can be explained by local stability analysis. In sharp contrast to previously reported results, cooperation is promoted by the spatial structure in the MBSG. Interestingly, we found that the frequency of cooperation of the MBSG on a scale-free network peaks at a specific value of the payoff parameter. This phenomenon indicates that properly encouraging selfish behaviors can optimally enhance the cooperation. The memory effects of individuals are discussed in detail and some nonmonotonous phenomena are observed on both lattices and scale-free networks. Our work may shed some new light on the study of evolutionary games over networks.

We develop a full theoretical approach to clustering in complex networks. A key concept is introduced, the edge multiplicity, that measures the number of triangles passing through an edge. This quantity extends the clustering coefficient in that it involves the properties of two—and not just one—vertices. The formalism is completed with the definition of a three-vertex correlation function, which is the fundamental quantity describing the properties of clustered networks. The formalism suggests different metrics that are able to thoroughly characterize transitive relations. A rigorous analysis of several real networks, which makes use of this formalism and the metrics, is also provided. It is also found that clustered networks can be classified into two main groups: the weak and the strong transitivity classes. In the first class, edge multiplicity is small, with triangles being disjoint. In the second class, edge multiplicity is high and so triangles share many edges. As we shall see in the following paper, the class a network belongs to has strong implications in its percolation properties.

The percolation properties of clustered networks are analyzed in detail. In the case of weak clustering, we present an analytical approach that allows us to find the critical threshold and the size of the giant component. Numerical simulations confirm the accuracy of our results. In more general terms, we show that weak clustering hinders the onset of the giant component whereas strong clustering favors its appearance. This is a direct consequence of the differences in the $k$-core structure of the networks, which are found to be totally different depending on the level of clustering. An empirical analysis of a real social network confirms our predictions.

We investigate the complex flow field around a sphere rising in a Maxwell fluid by means of the lattice Boltzmann simulation to provide insights into the strange negative wake experimentally observed behind a bubble or particle in non Newtonian fluids. The influence of the rise velocity, sphere diameter, and fluid’s rheology is considered through two dimensionless numbers: the Deborah number De and the Reynolds number Re. Our simulation shows that the negative wake appears behind the sphere when $\mathrm{De}>2$. On the other hand, the shape of the negative wake described by the opening angle $\theta$ of the upward flow cone surrounding the negative wake is mainly determined by the Reynolds number Re. These results reveal that the physical origin of the negative wake stems mainly from the competition between the elastic and viscous stresses in the fluid.

Localized regions of intense large-scale radiofrequency field are known to act like effective (“ponderomotive”) potential barriers, which scatter particles elastically and in the direction determined by the particle initial velocity rather than phase. In smaller-scale fields, transmission through a ponderomotive barrier is probabilistic and resembles tunneling of a quantum particle through a static potential. We derive asymptotic expressions for the phase-averaged transmission coefficient $T$ as a function of the particle energy ${\mathcal{E}}_0$. We show that, unlike for a truly quantum particle, $T\left({\mathcal{E}}_0\right)$ is of algebraic form and has a threshold, below which transmission does not occur. We also find a threshold in ${\mathcal{E}}_0$, above which all particles are transmitted regardless of their initial phase.

We experimentally generate different types of two-dimensional self-trapped photonic lattices in a photorefractive medium and analyze the induced refractive index change using two different methods. One method gives the first experimental Fourier space analysis of both linear and nonlinear self-trapped photonic lattices with periodic phase modulation using partially spatially incoherent multiband excitation of the lattice modes. The other method utilizes the waveguiding properties of the lattice to achieve a real space analysis of the induced refractive index change. The results of both methods are compared.

Various ensembles of random matrices with independent entries are analyzed by the replica formalism in the large-$N$ limit. A result on the Laplacian random matrix with Wigner-rescaling is generalized to arbitrary probability distribution.

A simple spin-flip process is analyzed under the presence of two heat reservoirs. While one flip process is triggered by a bath at temperature $T$, the inverse process is activated by a bath at a different temperature $T^{\prime }$. The situation can be described by using a master equation approach in a second quantized Hamiltonian formulation. The stationary solution leads to a generalized Fermi-Dirac distribution with an effective temperature $T_e$. Likewise the relaxation time is given in terms of $T_e$. Introducing a spin representation we perform a Landau expansion for the averaged spin $⟨\sigma ⟩$ as order parameter and consequently, a free energy functional can be derived. Owing to the two reservoirs the model is invariant with respect to a simultaneous change $\sigma \leftrightarrow -\sigma$ and $T\leftrightarrow T^{\prime }$. This symmetry generates a third order term in the free energy which gives rise a dynamically induced first order transition.

Ideal (monodomain) smectic-$A$ elastomers cross-linked in the smectic-$A$ phase are simply uniaxial rubbers, provided deformations are small. From these materials smectic-$C$ elastomers are produced by a cooling through the smectic-$A$ to smectic-$C$ phase transition. At least in principle, biaxial smectic elastomers could also be produced via cooling from the smectic-$A$ to a biaxial smectic phase. These phase transitions, respectively, from $D_{\infty h}$ to $C_{2h}$ and from $D_{\infty h}$ to $D_{2h}$ symmetry, spontaneously break the rotational symmetry in the smectic planes. We study the above transitions and the elasticity of the smectic-$C$ and biaxial phases in three different but related models: Landau-like phenomenological models as functions of the Cauchy-Saint-Laurent strain tensor for both the biaxial and the smectic-$C$ phases and a detailed model, including contributions from the elastic network, smectic layer compression, and smectic-$C$ tilt for the smectic-$C$ phase as a function of both strain and the $c$-director. We show that the emergent phases exhibit soft elasticity characterized by the vanishing of certain elastic moduli. We analyze in some detail the role of spontaneous symmetry breaking as the origin of soft elasticity and we discuss different manifestations of softness like the absence of restoring forces under certain shears and extensional strains.

We present experimental evidence of the presence of isolated screw dislocations in smectic-$A$ liquid crystals observed by polarizing microscopy. In a wedge-shaped homeotropic cell, the edge and screw dislocations interaction gives rise to a strong-enough optical contrast and makes visible their mutual intersections at temperatures close to the smectic-$A$ to smectic-$C$ phase transition temperature. The nature of the defects is confirmed by confocal microscopy observations. At large scale we observe a forest of screw dislocations, perpendicular to the smectic layers, across the thickness of the cell (end-on configuration). Their density varies between ${10}^9$ and ${10}^{12}{\mathrm{m}}^{-2}$. In situ observations of dislocations under stress, in the optical microscope, provide quantitative information about the screw-edge dislocation interactions. The latter interaction is calculated in the unharmonic approximation and it gives rise to an observed yield stress.

The distribution of inclusion-rich domains in membranes with active two-state inclusions is studied by simulations. Our study shows that typical size of inclusion-rich domains $\left(L\right)$ can be controlled by inclusion activities in several ways. When there is effective attraction between state-1 inclusions, we find: (i) Small domains with only several inclusions are observed for inclusions with time scales $(\sim {10}^{-3}\mathrm{s})$ and interaction energy $[\sim \mathcal{O}\left(k_{\mathrm{B}}T\right)]$ comparable to motor proteins. (ii) $L$ scales as $1∕3$ power of the lifetime of state-1 for a wide range of parameters. (iii) $L$ shows a switch-like dependence on state-2 lifetime $k_{12}^{-1}$. That is, $L$ depends weakly on $k_{12}$ when $k_{12}<k_{12}^{*}$ but increases rapidly with $k_{12}$ when $k_{12}>k_{12}^{*}$, the crossover $k_{12}^{*}$ occurs when the diffusion length of a typical state-2 inclusion within its lifetime is comparable to $L$. (iv) Inclusion-curvature coupling provides another length scale that competes with the effects of transition rates.

We study how decoherence rules the quantum-classical transition of the kicked harmonic oscillator. The system presents classical dynamics that ranges from regular to strong chaotic behavior depending on the amplitude of the kicks. We show that for regular and mixed classical dynamics, and in the presence of noise, the distance between the classical and quantum phase space distributions is proportional to a single parameter $\chi \equiv K{\hslash }_{\mathrm{eff}}^2∕4D^{3∕2}$, which relates the effective Planck constant, ${\hslash }_{\mathrm{eff}}$, to the kicking strength, $K$, and the diffusion constant, $D$. This relation between classical and quantum distributions is valid when $\chi <1$, a case that is always attainable in the semiclassical regime, independent of the value of the strength of noise given by $D$. Our results extend a recent study performed in the chaotic regime.

A coupling function that describes the interaction between self-sustained oscillators in a phase equation is derived and applied experimentally to Belousov-Zhabotinsky (BZ) oscillators. It is demonstrated that the synchronous behavior of coupled BZ reactors is explained extremely well in terms of the coupling function thus obtained. This method does not require comprehensive knowledge of either the oscillation mechanism or the interaction among the oscillators, both of these being often difficult to elucidate in an actual system. These facts enable us to accurately analyze the weakly coupled entrainment phenomenon through the direct measurement of the coupling function.

Diffraction gratings are elementary tools for much of optics and spectroscopy. Here, at microwave frequencies, we provide a new perspective on these fundamental structures. A transmission diffraction grating is presented that has diffracted beams emanating from one surface only. It can thus function either as a transmission grating with no reflected orders (other than zero) or, in the reverse configuration, as a partially transmitting structure with diffracted orders in reflection only.

In this paper, a wire mesh metallic photonic crystal is modelized as a stack of gratings of period $d$ made of very thin infinitely metallic rods (the conductivity is assumed to be infinite) of radius $a$ $(a⪡d)$. This structure is illuminated by a $s$-polarized plane wave of wavelength $\lambda$. First, we give the diffracted field in closed form for very large wavelengths compared to the period $(\lambda ⪢d)$. We derive a very accurate formula for the cut wavelength and we show that the plasma frequency in wire photonic crystals depends upon the Bloch vector. This latest formula is checked numerically.

We investigate the dependence of the stickiness parameters $t_{ij}=1∕\left(12{\tau }_{ij}\right)$—where the ${\tau }_{ij}$ are the conventional Baxter parameters—on the solute diameters ${\sigma }_i$ and ${\sigma }_j$ in multicomponent sticky hard sphere (SHS) models for fluid mixtures of mesoscopic neutral particles. A variety of simple but realistic interaction potentials, utilized in the literature to model short-ranged attractions present in real solutions of colloids or reverse micelles, is reviewed. We consider: (i) van der Waals attractions, (ii) hard-sphere-depletion forces, (iii) polymer-coated colloids, and (iv) solvation effects (in particular hydrophobic bonding and attractions between reverse micelles of water-in-oil microemulsions). We map each of these potentials onto an equivalent SHS model by requiring the equality of the second virial coefficients. The main finding is that, for most of the potentials considered, the size-dependence of $t_{ij}(T,{\sigma }_i,{\sigma }_j)$ can be approximated by essentially the same expression, i.e., a simple polynomial in the variable ${\sigma }_i{\sigma }_j∕{\sigma }_{ij}^2$, with coefficients depending on the temperature $T$, or—for depletion interactions—on the packing fraction ${\eta }_0$ of the depletant particles.

We report on theoretical results concerning the relation between the liquid-liquid transition and the density anomaly for a family of ramp potentials (hard-core plus linear short range repulsion and linear long range attraction). Using first order perturbation, we have studied the influence of the range of the attractive interactions, taking the repulsive part of the interaction as the reference system. Two different mechanisms of liquid-liquid coexistence have been predicted: attraction and compression. The attractive case is attributed to long ranged potentials, while the second one is obtained when the interaction is shortened. The density anomaly appears linked to regions where the temperature derivative of the density derivative of the energy is bigger (in absolute value) than a limit. This condition is fulfilled when the range of the attractive part of the potential is short enough.

We investigate the magnetostriction of field-structured magnetoelastomers, which are an important class of materials that have great potential as both sensors and actuators. Field-structured magnetoelastomers are synthesized by suspending magnetic particles in a polymeric resin and subjecting these to magnetic structuring fields during polymerization. These structuring fields can consist of as many as three orthogonal ac components, allowing a wide variety of particles structures—chains, sheets, or networks—to be formed. A principal issue is how particle structure and loading affects the magnetostriction of these materials. To investigate magnetostriction in these field-structured composites we have constructed a constant stress, optical cantilever apparatus capable of $1\mathrm{ppm}$ strain resolution. Magnetoelastomers having a wide range of particle loadings and structures are investigated, and it is shown that the observed deformation depends strongly on composite structure. The best magnetoelastomers exhibit a contractive strain of $10000\mathrm{ppm}$, the worst materials exhibit a negative, tensile response, which we show is due to the dominance of demagnetizing field effects over magnetostriction. Finally, some discussion is given to the surprising finding that magnetostriction is proportional to the sample prestrain. Simulations of a chain of particles in an elastomer show that particle clumping transitions can occur, but this does not account for the dependence of magnetostriction on prestrain.

Dielectric measurements under steady shear flow are made to clarify the shear induced structural changes in the nematic $\left(N\right)$ and the smectic-$A$ $\left(S_A\right)$ phases of 4-n-octyl-$4^{\prime }$-cyanobiphenyl (8CB). In the $N$ phase, the temperature dependence of the dielectric permittivity shows a characteristic behavior, which can be consistently interpreted based on the shear-induced structures suggested by x-ray scattering; flow alignment of the director and some dynamical structures characterized by different precessional motions of the director. For the shear-induced structural change in the $S_A$ phase, it is suggested that undulation instability would play an important role.

We conducted structural characterizations, reflection measurements, and theoretical simulations on the iridescent green and purple neck feathers of domestic pigeons (Columba livia domestica). We found that both green and purple barbules are composed of an outer keratin cortex layer surrounding a medullary layer. The thickness of the keratin cortex layer shows a distinct difference between green and purple barbules. Green barbules vary colors from green to purple with the observing angle changed from normal to oblique, while purple barbules from purple to green in an opposite way. Both the experimental and theoretical results suggest that structural colors in green and purple neck feathers should originate from the interference in the top keratin cortex layer, while the structure beyond acts as a poor mirror.