We propose a new type of waveforms in two-dimensional (2D) and three-dimensional (3D) discrete media-multilegged extended nonlinear structures (ENSs), built as arrays of lattice solitons (tiles and stones, in the 2D and 3D cases, respectively). We study the stability of the tiles and stones analytically, and then extend them numerically to complete ENS forms for both 2D and 3D lattices, aiming to single out stable ENSs. The predicted patterns can be realized in Bose-Einstein condensates trapped in deep optical lattices, crystals built of microresonators, and 2D photonic crystals. In the latter case, the patterns provide for a technique for writing reconfigurable virtual partitions in multipurpose photonic devices.

We investigate precursors and the predictability of extreme increments in a time series. The events we are focusing on consist in large increments within successive time steps. We are especially interested in understanding how the quality of the predictions depends on the strategy to choose precursors, on the size of the event, and on the correlation strength. We study the prediction of extreme increments analytically in an autoregressive process of order 1, and numerically in wind speed recordings and long-range correlated autoregressive moving average processes data. We evaluate the success of predictions via receiver-operator characteristics (ROC curves). Furthermore, we observe an increase of the quality of predictions with increasing event size and with decreasing correlation in all examples. Both effects can be understood by using the likelihood ratio as a summary index for smooth ROC curves.

Given an unconditionally stable algorithm for solving the Cahn-Hilliard equation, we present a general calculation for an analytic time step $\mathrm{\Delta }\tau$ in terms of an algorithmic time step $\mathrm{\Delta }t$. By studying the accumulative multistep error in Fourier space and controlling the error with arbitrary accuracy, we determine an improved driving scheme $\mathrm{\Delta }t=A{t}^{2∕3}$ and confirm the numerical results observed in a previous study [Cheng and Rutenberg, Phys. Rev. E 72, 055701(R) (2005)].

The fact that the elastic constant for bending a layer of smectic-$C$ liquid crystal along its $c$ director differs from the value for bending in the perpendicular direction has recently been shown to give rise to interactions between distant layers. The effect of this entropy-induced interaction is to favor a parallel or antiparallel alignment of the $c$ directors in these nonadjacent layers. We calculate in detail the range and strength of this interaction in both infinite and finite samples, and find the results to depend mainly on the ratio of the average layer bending elastic constant to the layer compression modulus. At low values of this ratio, the interlayer interaction is of long range in a bulk sample, while at high values of the ratio it decays as the inverse cube of the interlayer distance. For a sample confined between rigid substrates parallel to the layers, the interaction is greatly reduced. For a free-standing film the interaction may be enhanced if the surface tension is weak, but may be diminished if the surface tension is strong.

The polymorphism of a new organosiloxane tetrapode compound with cyano terminal polar groups was characterized by means of polarizing optical microscopy and x-ray diffraction. The compound exhibits smectic-$A$ and smectic-$C$ phases with a partial bilayer arrangement due to a certain degree of head-to-head association of the mesogenic units through their cyano end groups. On the basis of x-ray diffraction results, evidencing the microsegregation of polyphilic molecules, packing models for the smectic-$A$ and smectic-$C$ phases are proposed. A high degree of smectic positional order and a relatively low value of the tilt angle in the smectic-$C$ phase are indicated. Molecular dynamics of the studied compound was investigated by means of proton NMR relaxometry. The frequency dispersions of the spin-lattice relaxation time $\left(T_1\right)$ show that the relaxation is induced by three rotational modes of individual dendrimer arms with frequencies between ${10}^6$ and ${10}^9\mathrm{Hz}$. In the smectic phases, the effect of individual rotations is overwhelmed by a well expressed contribution of layer undulations at Larmor frequencies below $\sim 10\mathrm{MHz}$. The appearance of this relaxation mechanism over the frequency range of three decades is so far unique in the case of thermotropic liquid crystals. The analysis of the layer undulations contribution supports the microsegregation model of the smectic phases by revealing a slowing-down of translational diffusion and the lack of interactions among the sublayers formed by the mesogenic groups.

Dielectric spectroscopy measurements were performed for aqueous solutions of short single-stranded DNA with 30 to 120 bases of thymine over a frequency range of ${10}^5$ to ${10}^8\mathrm{Hz}$. Dielectric dispersion was found to include two relaxation processes in the ranges from ${10}^5$ to ${10}^6$ and from ${10}^6$ to ${10}^8\mathrm{Hz}$, respectively, with the latter mainly discussed in this study. The dielectric increment and the relaxation time of the high-frequency relaxation of DNA in solutions without added salt exhibited concentration and polymer-length dependences eventually identical to those for dilute polyion solutions described in previous studies. For solutions with added salt, on the other hand, those dielectric parameters were independent of salt concentration up to a certain critical value and started to decrease with further increasing salt concentration. This critical behavior is well explained by our newly extended cell model that takes into account the spatial distribution of loosely bound counterions around DNA molecules as a function of salt concentration.

The hippocampus has the capacity for reactivating recently acquired memories and it is hypothesized that one of the functions of sleep reactivation is the facilitation of consolidation of novel memory traces. The dynamic and network processes underlying such a reactivation remain, however, unknown. We show that such a reactivation characterized by local, self-sustained activity of a network region may be an inherent property of the recurrent excitatory-inhibitory network with a heterogeneous structure. The entry into the reactivation phase is mediated through a physiologically feasible regulation of global excitability and external input sources, while the reactivated component of the network is formed through induced network heterogeneities during learning. We show that structural changes needed for robust reactivation of a given network region are well within known physiological parameters.

We derive the equation of state of DNA under tension that features a loop. Such loops occur transiently during DNA condensation in the presence of multivalent ions or permanently through sliding protein linkers such as condensin. The force-extension relation of such looped-DNA modeled as a wormlike chain is calculated via path integration in the semiclassical limit. This allows us to rigorously determine the high stretching asymptotics. Notably the functional form of the force-extension curve resembles that of straight DNA, yet with a strongly renormalized apparent persistence length. We also present analogous results for DNA under tension with several protein-induced kinks and/or loops. That means that the experimentally extracted single-molecule elasticity does not necessarily only reflect the bare DNA stiffness, but can also contain additional contributions that depend on the overall chain conformation and length.

Premixed methane-air mixture is introduced at the center of two parallel circular quartz plates separated by a millimeter scale distance $(⩽5\mathrm{mm})$. Two plates are heated with an external heater to create a positive temperature gradient condition along the flow direction. The combustion characteristics of lean methane-air mixtures are investigated at $\varphi =0.67$ and $\varphi =0.85$ in the laminar flow regime $(\mathrm{Re}\sim 500–1800)$. Contrary to the general perception of a stable premixed flame front at a radial location, a large variety of unstable and dynamic flame front patterns are observed at $\varphi =0.67$. At $\varphi =0.85$, stable flame propagation mode dominated the regime diagram and unstable flame propagation modes are observed under limited conditions. In these unstable flame propagation modes, single and multiple flame fronts rotate around the center at a rate of $15–50\mathrm{Hz}$.

We have explored flow states of the Taylor-Couette system in the high radius ratio limit (0.99) with only the inner cylinder rotating. The usual Taylor vortices and wavy vortices were found, but the transition to higher states is interrupted by the emergence of intermittent short-wavelength bursts. This flow state is distinct from previously observed bursting phenomena found at lower radius ratios, while it is similar in appearance to states found in inclined layer convection.

In this paper we continue the analytical study of the sabra shell model of energy turbulent cascade. We prove the global existence of weak solutions of the inviscid sabra shell model, and show that these solutions are unique for some short interval of time. In addition, we prove that the solutions conserve energy, provided that the components of the solution satisfy $\mid u_n\mid \le C{k}_n^{-1∕3}\left[\sqrt{n}\log \right(n+1){]}^{-1}$ for some positive absolute constant $C$, which is the analog of the Onsager’s conjecture for the Euler’s equations. Moreover, we give a Beal-Kato-Majda type criterion for the blow-up of solutions of the inviscid sabra shell model and show the global regularity of the solutions in the “two-dimensional” parameters regime.

We point out a problem in the parametric estimation method based on the adaptive-feedback algorithm proposed by D. Huang [Phys. Rev. E 73, 066204 (2006)] both illustratively and analytically. Furthermore, under some hypotheses, we prove that it is not the so-called chaotic dynamical characteristic but the linear independence between the right-hand functions and the estimated parameters of the intrinsic system in its attractor that counts in the availability of this algorithm in estimating the unknown parameters.

The dynamical behavior of a system consisting of a (heavy) piston and two rectangular boxes, each containing two hard disks and in contact with the piston, is studied based on a projection operator method for a microcanonical ensemble. We derive a coupled set of nonlinear equations for slow variables of the system and solve it to confirm that our theory with no adjustable parameters reproduces experimental results fairly well. Some limitations of the theory are discussed from the viewpoint of the separation of slow and fast time scales and ergodicity.

We have investigated the random walk problem in a finite system and studied the crossover induced in the persistence probability by the system size. Analytical and numerical work show that the scaling function is an exponentially decaying function. We consider two cases of trapping, one by a box of size $L$ and the other by a harmonic trap. Our analytic calculations are supported by numerical works. We also present numerical results on the harmonically trapped randomly accelerated particle and the randomly accelerated particle with viscous drag.

A generic feature of systems with long-range interactions is the presence of quasistationary states with non-Gaussian single particle velocity distributions. For the case of the Hamiltonian mean-field model, we demonstrate that a maximum entropy principle applied to the associated Vlasov equation explains known features of such states for a wide range of initial conditions. We are able to reproduce velocity distribution functions with an analytic expression which is derived from the theory with no adjustable parameters. A normal diffusion of angles is detected, which is consistent with Gaussian tails of velocity distributions. A dynamical effect, two oscillating clusters surrounded by a halo, is also found and theoretically justified.

We have calculated the equilibrium shape of the axially symmetric Plateau border (PB) along which a spherical bubble contacts a flat wall, by numerical integration of Laplace’s equation. We found that the (spherical) film prolongation into the PB meets the wall at an internal angle $\varphi \le \pi ∕2$; the deviation $\mathrm{\Delta }\varphi \equiv \pi ∕2-\varphi$ is an increasing function of the liquid fraction and of the liquid-wall contact angle. For the equivalent dry bubble (i.e., no PB) this deviation can be accounted for in terms of a negative line tension $\tau$ associated with the PB, and which can be determined from $\mathrm{\Delta }\varphi$. We have also calculated the (negative) PB excess energy $ϵ$, defined as the energy per unit length of the PB’s liquid-gas and liquid-wall interfaces minus that of the film prolongation into the PB and of the dry wall. For $A^{1∕2}∕x_I\lesssim 0.4$ (where $A$ is the PB cross-sectional area and $x_I$ is the radius of the extrapolated contact line), it was found that $\mid ϵ\mid \propto A^{1∕2}$. Finally, we derived a general relationship involving $ϵ$, $\tau$ and $A$ which yields $\tau =ϵ∕2$ when $\mid ϵ\mid \propto A^{1∕2}$, i.e., for not too large PBs; larger PBs have $\mid \tau \mid <\mid ϵ\mid ∕2$.

We calculate the effective fluctuation-induced force between spherical or disklike colloids trapped at a flat, fluid interface mediated by thermally excited capillary waves. This Casimir-type force is determined by the partition function of the system which in turn is calculated in a functional integral approach, where the restrictions on the capillary waves imposed by the colloids are incorporated by auxiliary fields. In the long-range regime the fluctuation-induced force is shown to depend sensitively on the boundary conditions imposed at the three-phase contact line between the colloids and the two fluid phases. Separating the colloid fluctuations from the fluctuations of the capillary wave field leads to competing repulsive and attractive contributions, respectively, which give rise to cancellations of the leading terms. In a second approach based on a multipole expansion of the Casimir interaction, these cancellations can be understood from the vanishing of certain multipole moments enforced by the boundary conditions. We also discuss the connection of the different types of boundary conditions to certain external fields acting on the colloids which appear to be realizable by experimental techniques such as the laser tweezer method.

Twist-grain-boundary phases in smectics are the geometrical analogs of the Abrikosov flux lattice in superconductors. At large twist angles, the nonlinear elasticity is important in evaluating their energetics. We analytically construct the height function of a $\pi ∕2$ twist-grain-boundary phase in smectic-A liquid crystals, known as Schnerk’s first surface. This construction, utilizing elliptic functions, allows us to compute the energy of the structure analytically. By identifying a set of heretofore unknown defects along the pitch axis of the structure, we study the necessary topological structure of grain boundaries at other angles, concluding that there exist a set of privileged angles and that the $\pi ∕2$ and $\pi ∕3$ grain boundary structures are particularly simple.

We experimentally and numerically demonstrate the dual synchronization of chaos in two pairs of one-way-coupled Mackey-Glass electronic circuits with time-delayed feedback. The outputs of the two drive circuits are mixed and used both for the feedback signal to the two drive circuits and for the transmission signal to the two response circuits. We investigate the regions for achieving dual synchronization of chaos when the delay time is mismatched between the drive and response circuits.

We report experiments on turbulent Rayleigh-Bénard convection of air in a cylindrical large-scale facility with a diameter of $7\text{meters}$ and Rayleigh numbers up to $\mathrm{Ra}\approx {10}^{12}$. The facility is used to explore the structure of the large-scale circulation for continuously varying aspect ratios between $\Gamma \approx 1$ and $\Gamma \approx 10$. Using autocorrelation functions derived from high-resolution time series of temperature and velocity near the cooling plate we demonstrate that the well-known single-roll structure (often called “wind”) breaks down when the aspect ratio increases beyond the critical value ${\Gamma }_1=1.68\pm 0.22$. We further show that at ${\Gamma }_2=3.66\pm 0.46$ a second transition from an oscillatory two-roll structure to an unstable multi-roll structure takes place. The value of ${\Gamma }_2$ represents a lower bound for the aspect ratio that is necessary to reach homogeneous convection—a turbulent state that is free from the influence of lateral walls.