A general high-order finite element method for solving partial differential equations, called the COnstraints Oriented Library (COOL) method, is presented. This $hp$ approach takes into account the underlying nature of the corresponding physical problem, and thus avoids the generation of nonphysical solutions. In the COOL method, all terms in a variational form are represented by the same functional dependence and by the same regularity, thus eliminating regularity constraints imposed by standard numerical methods. External constraints, such as the incompressibility condition appearing in the Maxwell or Navier-Stokes equations, can then be satisfied identically and are eliminated algebraically. This reduces the number of variables, and leads to well-conditioned matrix problems. The consequence is that only physically relevant solutions remain. The COOL method also satisfies automatically internal constraints, such as occur for the grad(div) and curl(curl) operators, and this for any geometry. This approach can be applied to a wide range of physical problems, including fluid flows, electromagnetics, material sciences, ideal linear magnetohydrodynamic stability analysis, and Alfvén wave heating of fusion plasmas. Results obtained by applying the COOL method to the grad(div) and curl(curl) operators, the Stokes problem, and the steady and unsteady Navier-Stokes equations are presented.

The characteristic function of the work performed by an external time-dependent force on a Hamiltonian quantum system is identified with the time-ordered correlation function of the exponentiated system’s Hamiltonian. A similar expression is obtained for the averaged exponential work which is related to the free energy difference of equilibrium systems by the Jarzynski work theorem.

Dispersions of single walled carbon nanotubes (SWNTs), with an effective aspect ratio of $\sim 650$, in poly(ethylene oxide) (PEO) form fractal superstructures above their geometrical percolation and display rheological properties that follow time-temperature-composition-strain superpositioning. The concentration dependences of the elastic modulus of the network and the onset-strain for shear thinning are consistent with the short-range nature of the interactions that dominate these dispersions. The strain dependence of the damping behavior for the nanocomposites shows concentration invariance when represented against the local strain experienced by the network element, with the onset occurring at a local strain value of 0.1, similar to other nanocomposite systems dominated by weak interactions.

We show that using local chemical stimulations it is possible to imprint persisting (days) multiple memories (collective modes of neuron firing) in the activity of cultured neural networks. Microdroplets of inhibitory antagonist are injected at a location selected based on real-time analysis of the recorded activity. The neurons at the stimulated locations turn into a focus for initiating synchronized bursting events (the collective modes) each with its own specific spatiotemporal pattern of neuron firing.

We introduce transverse ferromagnetic interactions, in addition to a simple transverse field, to accelerate the convergence of quantum annealing of the random-field Ising model. The conventional approach using only the transverse-field term is known to be plagued by slow convergence when the true ground state has strong ferromagnetic characteristics for the random-field Ising model. The transverse ferromagnetic interactions are shown to improve the performance significantly in such cases. This conclusion is drawn from the analyses of the energy eigenvalues of instantaneous stationary states as well as by the very fast algorithm of Bethe-type mean-field annealing adopted to quantum systems. The present study highlights the importance of a flexible choice of the type of quantum fluctuations to achieve the best possible performance in quantum annealing. The existence of such flexibility is an outstanding advantage of quantum annealing over simulated annealing.

We study nematic liquid crystal configurations in a prototype bistable device—the post aligned bistable nematic (PABN) cell. Working within the Oseen-Frank continuum model, we describe the liquid crystal configuration by a unit-vector field $\mathbf{n}$, in a model version of the PABN cell. First, we identify four distinct topologies in this geometry. We explicitly construct trial configurations with these topologies which are used as initial conditions for a numerical solver, based on the finite-element method. The morphologies and energetics of the corresponding numerical solutions qualitatively agree with experimental observations and suggest a topological mechanism for bistability in the PABN cell geometry.

The properties of certain networks are determined by hidden variables that are not explicitly measured. The conditional probability (propagator) that a vertex with a given value of the hidden variable is connected to $k$ other vertices determines all measurable properties. We study hidden variable models and find an averaging approximation that enables us to obtain a general analytical result for the propagator. Analytic results showing the validity of the approximation are obtained. We apply hidden variable models to protein-protein interaction networks (PINs) in which the hidden variable is the association free energy, determined by distributions that depend on biochemistry and evolution. We compute degree distributions as well as clustering coefficients of several PINs of different species; good agreement with measured data is obtained. For the human interactome two different parameter sets give the same degree distributions, but the computed clustering coefficients differ by a factor of about 2. This shows that degree distributions are not sufficient to determine the properties of PINs.

We consider scattering of stochastic scalar fields on deterministic as well as on random media, occupying a finite domain. The scattering is characterized by a generalized scattering matrix which transforms the angular correlation function of the incident field into the angular correlation function of the scattered field. Within the accuracy of the first Born approximation this matrix can be expressed in a simple manner in terms of the scattering potential of the scatterer. Apart from determining the angular distribution of the spectral intensity of the scattered field, the scattering matrix makes it possible also to determine the changes in the state of coherence of the field produced on scattering.

It was shown some years ago that the spectrum of a stochastic scalar field depends not only on the source spectrum but also on the degree of coherence of the source. In this paper we show that there are electromagnetic fields for which not only the state of coherence of the source, but also its degree of polarization affect the spectrum of the radiated field. We illustrate the analysis by diagrams which show the far-zone spectra of some stochastic electromagnetic beams generated by sources of different states of coherence and different degrees of polarization. The spectra of the radiated field depend both on coherence properties of the source and its degree of polarization and are found to be different in different directions of observation.

We study biased, diffusive transport of Brownian particles through narrow, spatially periodic structures in which the motion is constrained in lateral directions. The problem is analyzed under the perspective of the Fick-Jacobs equation, which accounts for the effect of the lateral confinement by introducing an entropic barrier in a one-dimensional diffusion. The validity of this approximation, based on the assumption of an instantaneous equilibration of the particle distribution in the cross section of the structure, is analyzed by comparing the different time scales that characterize the problem. A validity criterion is established in terms of the shape of the structure and of the applied force. It is analytically corroborated and verified by numerical simulations that the critical value of the force up to which this description holds true scales as the square of the periodicity of the structure. The criterion can be visualized by means of a diagram representing the regions where the Fick-Jacobs description becomes inaccurate in terms of the scaled force versus the periodicity of the structure.

The dynamic structure factor $S(Q,\omega )$ of water has been determined by high-resolution inelastic x-ray scattering (IXS) in a momentum $\left(Q\right)$ and energy $\left(E\right)$ transfer range extending from $2\text{to}4{\mathrm{nm}}^{-1}$ and from $\pm 40\mathrm{meV}$. IXS spectra have been recorded along an isobaric path $\left(400\text{bar}\right)$ in a temperature $\left(T\right)$ interval ranging from ambient up to supercritical $(T>647\mathrm{K})$ conditions. The experimental data have been described in the frame of the generalized hydrodynamic theory, utilizing a model based on the memory function approach. This model allows identifying the active relaxation processes which affect the time decay of density fluctuations, as well as a direct determination of the $Q$, $T$, and density $\left(\rho \right)$ dependencies of the involved transport parameters. The experimental spectra are well described by considering three different relaxation processes: the thermal, the structural, and the instantaneous one. On approaching supercritical conditions, we observe that the microscopic mechanism responsible for the structural relaxation is no longer related to the making and breaking of intermolecular bonds, but to binary intermolecular collisions.

We study a schematic mode-coupling model in which the ideal glass transition is cut off by a decay of the quadratic coupling constant in the memory function. (Such a decay, on a time scale ${\tau }_I$, has been suggested as the likely consequence of activated processes.) If this decay is complete, so that only a linear coupling remains at late times, then the $\alpha$ relaxation shows a temporal crossover from a relaxation typical of the unmodified schematic model to a final strongly slower-than-exponential relaxation. This crossover, which differs somewhat in form from previous schematic models of the cutoff glass transition, resembles light-scattering experiments on colloidal systems, and can exhibit a “slower-than-$\alpha$” relaxation feature hinted at there. We also consider what happens when a similar but incomplete decay occurs, so that a significant level of quadratic coupling remains for $t⪢{\tau }_I$. In this case the correlator acquires a third, weaker relaxation mode at intermediate times. This empirically resembles the $\beta$ process seen in many molecular glass formers. It disappears when the initial as well as the final quadratic coupling lies on the liquid side of the glass transition, but remains present even when the final coupling is only just inside the liquid (so that the $\alpha$ relaxation time is finite, but too long to measure). Our results are suggestive of how, in a cutoff glass, the underlying “ideal” glass transition predicted by mode-coupling theory can remain detectable through qualitative features in dynamics.

We present a model for smectic elastomer membranes which includes elastic and liquid-crystalline degrees of freedom. Based on our model, we determined the qualitative phase diagram of a smectic elastomer membrane using mean-field theory. This phase diagram is found to comprise five phases, viz., smectic-$A$-flat, smectic-$A$-crumpled, smectic-$C$-flat, smectic-$C$-crumpled, and smectic-$C$-tubule phases, where in the latter phase, the membrane is flat in the direction of mesogenic tilt and crumpled in the perpendicular direction. The transitions between adjacent phases are second-order phase transitions. We study in some detail the elasticity of the smectic-$C$-flat and the smectic-$C$-tubule phases which are associated with a spontaneous breaking of in-plane rotational symmetry. As a consequence of the Goldstone theorem, these phases exhibit soft elasticity characterized by the vanishing of in-plane shear moduli.

The pearl necklace instability has been predicted to occur for single polyelectrolyte chains immersed in an incompatible (or poor) solvent, but still direct imaging of this chain conformation is not conclusive. We therefore examine how a surface may interact with the polyelectrolyte to, possibly, inhibit this instability. We show from explicit calculations that for strongly absorbed polyelectrolyte chains another conformation of the chain is stable in parts of the diagram of states. We support this calculation with more rigorous numerical, self-consistent mean field calculations which exhibit both elliptical globules and a pseudo pearl necklace structure.

The coexistence between different informational molecules has been the preferred mode to circumvent the limitation posed by imperfect replication on the amount of information stored by each of these molecules. Here we reexamine a classic package model in which distinct information carriers or templates are forced to coexist within vesicles, which in turn can proliferate freely through binary division. The combined dynamics of vesicles and templates is described by a multitype branching process which allows us to write equations for the average number of the different types of vesicles as well as for their extinction probabilities. The threshold phenomenon associated with the extinction of the vesicle population is studied quantitatively using finite-size scaling techniques. We conclude that the resultant coexistence is too frail in the presence of parasites and so confinement of templates in vesicles without an explicit mechanism of cooperation does not resolve the information crisis of prebiotic evolution.

The concept of controllability of linear systems from control theory is applied to networks inspired by biology. A node is in this context controllable if an external signal can be applied which can adjust the level (e.g., protein concentration) of the node in a finite time to an arbitrary value, regardless of the levels of the other nodes. The property of being downstream of the node to which the input is applied turns out to be a necessary but not a sufficient condition for being controllable. An interpretation of the controllability matrix, when applied to networks, is also given. Finally, two case studies are provided in order to better explain the concepts, as well as some results for a gene regulatory network of fission yeast.

A method is proposed for detecting from univariate data the presence of synchronization of a self-sustained oscillator by external driving with varying frequency. The method is based on the analysis of difference between the oscillator instantaneous phases calculated using continuous wavelet transform at time moments shifted by a certain constant value relative to each other. We apply our method to a driven asymmetric van der Pol oscillator, experimental data from a driven electronic oscillator with delayed feedback and human heartbeat time series. In the latest case, the analysis of the heart rate variability data reveals synchronous regimes between the respiration and slow oscillations in blood pressure.

We present a continuum-mechanical formulation and generalization of the Navier-Stokes $\alpha$ theory based on a general framework for fluid-dynamical theories with gradient dependencies. Our flow equation involves two additional problem-dependent length scales $\alpha$ and $\beta$. The first of these scales enters the theory through the internal kinetic energy, per unit mass, ${\alpha }^2{\mid \mathbf{D}\mid }^2$, where $\mathbf{D}$ is the symmetric part of the gradient of the filtered velocity. The remaining scale is associated with a dissipative hyperstress which depends linearly on the gradient of the filtered vorticity. When $\alpha$ and $\beta$ are equal, our flow equation reduces to the Navier-Stokes $\alpha$ equation. In contrast to the original derivation of the Navier-Stokes $\alpha$ equation, which relies on Lagrangian averaging, our formulation delivers boundary conditions. For a confined flow, our boundary conditions involve an additional length scale $\ell$ characteristic of the eddies found near walls. Based on a comparison with direct numerical simulations for fully developed turbulent flow in a rectangular channel of height $2h$, we find that $\alpha ∕\beta \sim {\mathrm{Re}}^{0.470}$ and $\ell ∕h\sim {\mathrm{Re}}^{-0.772}$, where Re is the Reynolds number. The first result, which arises as a consequence of identifying the internal kinetic energy with the turbulent kinetic energy, indicates that the choice $\alpha =\beta$ required to reduce our flow equation to the Navier-Stokes $\alpha$ equation is likely to be problematic. The second result evinces the classical scaling relation $\eta ∕L\sim {\mathrm{Re}}^{-3∕4}$ for the ratio of the Kolmogorov microscale $\eta$ to the integral length scale $L$. The numerical data also suggests that $\ell ⩽\beta$. We are therefore led to conjecture a tentative hierarchy, $\ell ⩽\beta <\alpha$, involving the three length scales entering our theory.

The amplitude and frequency dependence of dynamical mode locking phenomena in the dc- and ac-driven overdamped Frenkel-Kontorova model is studied by molecular-dynamics simulations. It was found that the Shapiro steps and the critical depinning force exhibit very complex behavior. The form of amplitude dependence is determined by the frequency of ac force, where the Bessel-type oscillations appear at the high frequencies. With a changing of frequency, after initial increase, the critical depinning force saturates, while the step width remains strongly frequency dependent even at the high frequencies. The dependence of frequency is strongly influenced by the amplitude of ac force where, in the large amplitude regime, the oscillations of the step width and the critical depinning force have been observed at the low frequencies. In the physical processes that stay behind amplitude and frequency oscillations of the step size, an analogy between the influence of amplitude and the period of the ac force is revealed. These oscillations are directly related to the existence and the stability of the interference phenomena in the real systems.