Single-molecule force spectroscopy using an atomic force microscope (AFM) can be used to measure the average unfolding force of proteins in a constant velocity experiment. In combination with Monte Carlo simulations and through the application of the Zhurkov-Bell model, information about the parameters describing the underlying unfolding energy landscape of the protein can be obtained. Using this approach, we have completed protein unfolding experiments on the polyprotein ${\left(I27\right)}_5$ over a range of pulling velocities. In agreement with previous work, we find that the observed number of protein unfolding events observed in each approach-retract cycle varies between one and five, due to the nature of the interactions between the polyprotein, the AFM tip, and the substrate, and there is an unequal unfolding probability distribution. We have developed a Monte Carlo simulation that incorporates the impact of this unequal unfolding probability distribution on the median unfolding force and the calculation of the protein unfolding energy landscape parameters. These results show that while there is a significant, unequal unfolding probability distribution, the unfolding energy landscape parameters obtained from use of the Zhurkov-Bell model are not greatly affected. This result is important because it demonstrates that the minimum acceptance criteria typically used in force extension experiments are justified and do not skew the calculation of the unfolding energy landscape parameters. We further validate this approach by determining the error in the energy landscape parameters for two extreme cases, and we provide suggestions for methods that can be employed to increase the level of accuracy in single-molecule experiments using polyproteins.

We study the problem of a particle or message that travels as a biased random walk towards a target node in a network in the presence of traps. The bias is represented as the probability $p$ of the particle to travel along the shortest path to the target node. The efficiency of the transmission process is expressed through the fraction $f_g$ of particles that succeed to reach the target without being trapped. By relating $f_g$ with the number $S$ of nodes visited before reaching the target, we first show that, for the unbiased random walk, $f_g$ is inversely proportional to both the concentration $c$ of traps and the size $N$ of the network. For the case of biased walks, a simple approximation of $S$ provides an analytical solution that describes well the behavior of $f_g$, especially for $p>0.5$. Also, it is shown that for a given value of the bias $p$, when the concentration of traps is less than a threshold value equal to the inverse of the mean first passage time (MFPT) between two randomly chosen nodes of the network, the efficiency of transmission is unaffected by the presence of traps and almost all the particles arrive at the target. As a consequence, for a given concentration of traps, we can estimate the minimum bias that is needed to have unaffected transmission, especially in the case of random regular (RR), Erdős-Rényi (ER) and scale-free (SF) networks, where an exact expression (RR and ER) or an upper bound (SF) of the MFPT is known analytically. We also study analytically and numerically, the fraction $f_g$ of particles that reach the target on SF networks, where a single trap is placed on the highest degree node. For the unbiased random walk, we find that $f_g\sim N^{-1/(\gamma -1)}$, where $\gamma$ is the power law exponent of the SF network.

We present a Hawkes-model approach to the foreign exchange market in which the high-frequency price dynamics is affected by a self-exciting mechanism and an exogenous component, generated by the pre-announced arrival of macroeconomic news. By focusing on time windows around the news announcement, we find that the model is able to capture the increase of trading activity after the news, both when the news has a sizable effect on volatility and when this effect is negligible, either because the news in not important or because the announcement is in line with the forecast by analysts. We extend the model by considering noncausal effects, due to the fact that the existence of the news (but not its content) is known by the market before the announcement.

Networks studied in many disciplines, including neuroscience and mathematical biology, have connectivity that may be stochastic about some underlying mean connectivity represented by a non-normal matrix. Furthermore, the stochasticity may not be independent and identically distributed (iid) across elements of the connectivity matrix. More generally, the problem of understanding the behavior of stochastic matrices with nontrivial mean structure and correlations arises in many settings. We address this by characterizing large random $N\times N$ matrices of the form $A=M+LJR$, where $M,L$, and $R$ are arbitrary deterministic matrices and $J$ is a random matrix of zero-mean iid elements. $M$ can be non-normal, and $L$ and $R$ allow correlations that have separable dependence on row and column indices. We first provide a general formula for the eigenvalue density of $A$. For $A$ non-normal, the eigenvalues do not suffice to specify the dynamics induced by $A$, so we also provide general formulas for the transient evolution of the magnitude of activity and frequency power spectrum in an $N$-dimensional linear dynamical system with a coupling matrix given by $A$. These quantities can also be thought of as characterizing the stability and the magnitude of the linear response of a nonlinear network to small perturbations about a fixed point. We derive these formulas and work them out analytically for some examples of $M,L$, and $R$ motivated by neurobiological models. We also argue that the persistence as $N\to \infty$ of a finite number of randomly distributed outlying eigenvalues outside the support of the eigenvalue density of $A$, as previously observed, arises in regions of the complex plane $\Omega$ where there are nonzero singular values of $L^{-1}(z\mathbf{1}-M)R^{-1}$ (for $z\in \Omega$) that vanish as $N\to \infty$. When such singular values do not exist and $L$ and $R$ are equal to the identity, there is a correspondence in the normalized Frobenius norm (but not in the operator norm) between the support of the spectrum of $A$ for $J$ of norm $\sigma$ and the $\sigma$ pseudospectrum of $M$.

It is common in the study of networks to investigate intermediate-sized (or “meso-scale”) features to try to gain an understanding of network structure and function. For example, numerous algorithms have been developed to try to identify “communities,” which are typically construed as sets of nodes with denser connections internally than with the remainder of a network. In this paper, we adopt a complementary perspective that communities are associated with bottlenecks of locally biased dynamical processes that begin at seed sets of nodes, and we employ several different community-identification procedures (using diffusion-based and geodesic-based dynamics) to investigate community quality as a function of community size. Using several empirical and synthetic networks, we identify several distinct scenarios for “size-resolved community structure” that can arise in real (and realistic) networks: (1) the best small groups of nodes can be better than the best large groups (for a given formulation of the idea of a good community); (2) the best small groups can have a quality that is comparable to the best medium-sized and large groups; and (3) the best small groups of nodes can be worse than the best large groups. As we discuss in detail, which of these three cases holds for a given network can make an enormous difference when investigating and making claims about network community structure, and it is important to take this into account to obtain reliable downstream conclusions. Depending on which scenario holds, one may or may not be able to successfully identify “good” communities in a given network (and good communities might not even exist for a given community quality measure), the manner in which different small communities fit together to form meso-scale network structures can be very different, and processes such as viral propagation and information diffusion can exhibit very different dynamics. In addition, our results suggest that, for many large realistic networks, the output of locally biased methods that focus on communities that are centered around a given seed node (or set of seed nodes) might have better conceptual grounding and greater practical utility than the output of global community-detection methods. They also illustrate structural properties that are important to consider in the development of better benchmark networks to test methods for community detection.

The propagation, in a shallow water, of nonlinear ring waves in the form of multisolitons is investigated theoretically. This is done by solving both analytically and numerically the cylindrical (also referred to as concentric) Korteweg-de Vries equation (cKdVE). The latter describes the propagation of weakly nonlinear and weakly dispersive ring waves in an incompressible, inviscid, and irrotational fluid. The spatiotemporal evolution is determined for a cylindrically symmetric response to the free fall of an initially given multisoliton ring. Analytically, localized solutions in the form of tilted solitons are found. They can be thought as single- or multiring solitons formed on a conic-modulated water surface, with an oblique asymptote in arbitrary radial direction (tilted boundary condition). Conversely, the ring solitons obtained from numerical solutions are localized single- or multiring structures (standard solitons), whose wings vanish along all radial directions (standard boundary conditions). It is found that the wave dynamics of these standard ring-type localized structures differs substantially from that of the tilted structures. A detailed analysis is performed to determine the main features of both multiring localized structures, particularly their break-up, multiplet formation, overlapping of pulses, overcoming of one pulse by another, “amplitude-width” complementarity, etc., that are typically ascribed to a solitonlike behavior. For all the localized structures investigated, the solitonlike character of the rings is found to be preserved during (almost) entire temporal evolution. Due to their cylindrical character, each ring belonging to one of these multiring localized structures experiences the physiological decay of the peak and the physiological increase of the width, respectively, while propagating (“amplitude-width” complementarity). As in the planar geometry, i.e., planar Korteweg-de Vries equation (pKdVE), we show that, in the case of the tilted analytical solutions, the instantaneous product $P=\left(\mathit{maximum}\mathit{amplitude}\right)\times {\left(\mathit{width}\right)}^2$ is rigorously constant during all the soliton spatiotemporal evolution. Nevertheless, in the case of the numerical solutions, we show that this product is not preserved; i.e., the instantaneous physiological variations of both peak and width of each ring do not compensate each other as in the tilted analytical case. In fact, the amplitude decay occurs faster than the width increase, so that $P$ decreases in time. This is more evident in the early times than in the asymptotic ones (where actually cKdVE reduces to pKdVE). This is in contrast to previous investigations on the early-time localized solutions of the cKdVE.

The dynamics of a particle confined in the elliptical stadium billiard with rectangular thickness $2t$, major axis $2a$, and minor axis $2b=2$ is numerically investigated in a reduced phase space with discrete time $n$. Both relative measure $r\left(n\right)$, with asymptotic value $r(n\to \infty )=r_{\infty }$ and Shannon entropy $s$, are calculated in the vicinity of a particular line in the $a\times t$ parameter space, namely $t_c=t_0\left(a\right)=\sqrt{a^2-1}$, with $a\in (1,\sqrt{4/3})$. If $t<t_c$, the billiard is known to exhibit a mixed phase space (regular and chaotic regions). As the line $t_c$ is crossed upwards by increasing $t$ with fixed $a$, we observe that the function $\psi \left(t\right)=\sqrt{1-r_{\infty }\left(t\right)}$ critically vanishes at $t=t_c$. In addition, we show that the function $c\left(t\right)=t(ds/dt)$ displays a pronounced peak at $t=t_c$. In the vicinity of $t_c$ ($t<t_c$), a chi-square tolerance of $1.0\times {10}^{-9}$ is reached when the numerically calculated functions $\psi \left(t\right)$ and $c\left(t\right)$ are fitted with renormalization group formulas with fixed parameters $\alpha =-0.0127,$ $\beta =0.34$, and $\Delta =0.5$. The results bear a remarkable resemblance to the famous $\lambda$ transition in liquid ${}^4\mathrm{He}$, where the two-component (superfluid and normal fluid) phase of He-II is critically separated from the fully entropic normal-fluid phase of He-I by the so-called $\lambda$ line in the pressure $\times$ temperature parameter space. The analogy adds support to a set of previous results by Markarian and coworkers, which indicate that the line $t_0\left(a\right)$ is a strong candidate for the bound for chaos in the elliptical stadium billiard if $a\in (1,\sqrt{4/3})$.

We study a two-component nonlinear Schrödinger system with equal, repulsive cubic interactions and different dispersion coefficients in the two components. We consider states that have a dark solitary wave in one component. Treating it as a frozen one, we explore the possibility of the formation of bright-solitonic structures in the other component. We identify bifurcation points at which such states emerge in the bright component in the linear limit and explore their continuation into the nonlinear regime. An additional analytically tractable limit is found to be that of vanishing dispersion of the bright component. We numerically identify regimes of potential stability, not only of the single-peak ground state (the dark-bright soliton), but also of excited states with one or more zero crossings in the bright component. When the states are identified as unstable, direct numerical simulations are used to investigate the outcome of the instability development. Although our principal focus is on the homogeneous setting, we also briefly touch upon the counterintuitive impact of the potential presence of a parabolic trap on the states of interest.

The electrically driven displacement of a viscous poorly conducting Newtonian fluid drop positioned between conducting parallel plates is studied both theoretically and experimentally. A mathematical expression for the average-steady velocity is developed by using Darcy flow analysis with an interface pressure that includes a contribution from Maxwell stresses at the advancing gas-liquid boundary. Experiments were performed using silicone oil at plate separation distances less than the capillary length for voltages ranging 250–750 V resulting in a change in velocity of approximately one order of magnitude. This suggested that the driving force was proportional to the square of the applied voltage, a feature that is common among fluid motion driven by electrical phenomena. The trailing interface revealed a disturbance for large fluid displacements that is analogous to a fingering instability. The channel widths were small such that we could predict the transition from a single finger that grows linearly to a single finger that grows exponentially and to multiple fingers that grow exponentially, all occurring within the range of voltages studied. The theory and experiments show good agreement with classic interfacial linear stability analysis.

In this paper we give the complete classification of solitons for a cubic nonlinear Schrödinger equation on the simplest network with a nontrivial topology: the tadpole graph, i.e., a ring with a half line attached to it and free boundary conditions at the junction. This is a step toward the modelization of condensate propagation and confinement in quasi-one-dimensional traps. The model, although simple, exhibits a surprisingly rich behavior and in particular we show that it admits: (i) a denumerable family of continuous branches of embedded solitons vanishing on the half line and bifurcating from linear eigenstates and threshold resonances of the system; (ii) a continuous branch of edge solitons bifurcating from the previous families at the threshold of the continuous spectrum with a pitchfork bifurcation; and (iii) a finite family of continuous branches of solitons without linear analog. All the solutions are explicitly constructed in terms of elliptic Jacobian functions. Moreover we show that families of nonlinear bound states of the above kind continue to exist in the presence of a uniform magnetic field orthogonal to the plane of the ring when a well definite flux quantization condition holds true. In this sense the magnetic field acts as a control parameter. Finally we highlight the role of resonances in the linearization as a signature of the occurrence of bifurcations of solitons from the continuous spectrum.

A random three-dimensional (3D) porous medium can be reconstructed from a two-dimensional (2D) image by reconstructing an image from the original 2D image, and then repeatedly using the result to reconstruct the next 2D image. The reconstructed images are then stacked together to generate the entire reconstructed 3D porous medium. To perform this successfully, a very important issue must be addressed, i.e., controlling the continuity and variability among adjacent layers. Continuity and variability, which are consistent with the statistics characteristic of the training image (TI), ensure that the reconstructed result matches the TI. By selecting the number and location of the sampling points in the sampling process, the continuity and variability can be controlled directly, and thus the characteristics of the reconstructed image can be controlled indirectly. In this paper, we propose and develop an original sampling method called three-step sampling. In our sampling method, sampling points are extracted successively from the center of $5\times 5$ and $3\times 3$ sampling templates and the edge area based on a two-point correlation function. The continuity and variability of adjacent layers were considered during the three steps of the sampling process. Our method was tested on a Berea sandstone sample, and the reconstructed result was compared with the original sample, using tests involving porosity distribution, the lineal path function, the autocorrelation function, the pore and throat size distributions, and two-phase flow relative permeabilities. The comparison indicates that many statistical characteristics of the reconstructed result match with the TI and the reference 3D medium perfectly.

In this paper a lattice Boltzmann equation (LBE) method is designed that is different from the previous LBE for the Cahn-Hilliard equation (CHE). The starting point of the present CHE LBE model is from the kinetic theory and the work of Lee and Liu [T. Lee and L. Liu, J. Comput. Phys. 229, 8045 (2010)]; however, because the CHE does not conserve the mass locally, a modified equilibrium density distribution function is introduced to treat the diffusion term in the CHE. Numerical simulations including layered Poiseuille flow, static droplet, and Rayleigh-Taylor instability have been conducted to validate the model. The results show that the predictions of the present LBE agree well with the analytical solution and other numerical results.

In classical statistical mechanics there is a clear correlation between relaxation to equilibrium and chaos. In contrast, for isolated quantum systems this relation is—to say the least—fuzzy. In this work we try to unveil the intricate relation between the relaxation process and the transition from integrability to chaos. We study the approach to equilibrium in two different many-body quantum systems that can be parametrically tuned from regular to chaotic. We show that a universal relation between relaxation and delocalization of the initial state in the perturbed basis can be established regardless of the chaotic nature of system.

Narrow escape problems consider the calculation of the mean first passage time (MFPT) for a particle undergoing Brownian motion in a domain with a boundary that is everywhere reflecting except for at finitely many small holes. Asymptotic methods for solving these problems involve finding approximations for the MFPT and average MFPT that increase in accuracy with decreasing hole sizes. While relatively much is known for the two-dimensional case, the results available for general three-dimensional domains are rather limited. This paper addresses the problem of finding the average MFPT for a class of three-dimensional domains bounded by the level surface of an orthogonal coordinate system. In particular, this class includes spheroids and other solids of revolution. The primary result presented is a two-term asymptotic expansion for the average MFPT of such domains containing an arbitrary number of holes. Steps are taken towards finding higher-order asymptotic expansions for both the average MFPT and the MFPT in these domains. The results for the average MFPT are compared to full numerical calculations performed with the comsol Multiphysics finite element solver for three distinct domains: prolate and oblate spheroids and biconcave disks. This comparison shows good agreement with the proposed two-term expansion of the average MFPT in the three domains.

The existence of a finite-size effect in one-dimensional oscillator systems causing the energy current power spectrum to saturate to a constant value at low frequencies is discussed. It is shown that a mode-coupling theory presented in earlier papers can be used to predict the frequency of onset of this finite-size effect. This can be used by researchers to plan simulations with large enough numbers of particles to avoid the presence of this finite-size effect.

The hydrodynamic part of the velocity autocorrelation function of a granular fluid in the homogeneous cooling state has been calculated by using mode-coupling theory for a finite system with periodic boundary conditions. The existence of the shearing instability, leading to a divergent behavior of the velocity flow fluctuations, is taken into account. A time region in which the velocity autocorrelation function exhibits a power-law decay, when time is measured by the number of collisions per particle, has been been identified. Also the explicit form of the exponential asymptotic long time decay has been obtained. The theoretical prediction for the power-law decay is compared with molecular dynamics simulation results, and a good agreement is found, after taking into account finite size corrections. The effects of approaching the shearing instability are also explored.

An effective method of controlling chaos is proposed. The control mechanism of the proposed method can be interpreted as the synchronization of a chaotic system with a system of harmonic oscillators. It is a practically noninvasive method that does not change the solution of dynamical systems, but stabilizes their periodic behavior by applying only small perturbations. The proposed method has fairly simple structure that suggests a clear stabilizing mechanism, and allows a straightforward evaluation of stability properties. This control scheme can be implemented experimentally even if the dynamical system is not analytically known. The efficiency of the proposed method is demonstrated through numerical simulation as well as experimental implementation.

We present a sparse matrix permutation from graph theory that gives stable incomplete lower-upper preconditioners necessary for iterative solutions to the steady-state density matrix for quantum optomechanical systems. This reordering is efficient, adding little overhead to the computation, and results in a marked reduction in both memory and runtime requirements compared to other solution methods, with performance gains increasing with system size. Either of these benchmarks can be tuned via the preconditioner accuracy and solution tolerance. This reordering optimizes the condition number of the approximate inverse and is the only method found to be stable at large Hilbert space dimensions. This allows for steady-state solutions to otherwise intractable quantum optomechanical systems.

Yagasaki et al. [Phys. Rev. E 89, 020301 (2014)] present results from a molecular dynamics trajectory illustrating coarsening of ice, which they interpret as evidence of transient coexistence between two distinct supercooled phases of liquid water. We point out that neither two distinct liquids nor criticality are demonstrated in this simulation study. Instead, the illustrated trajectory is consistent with coarsening behaviors analyzed and predicted in earlier work by others.

Two different scenarios have been proposed on the phase separation occurring in the deeply supercooled liquid water. We discuss what we can derive from our simulation results for the two scenarios and propose a way for future investigation. We also demonstrate that the phase separation in the supercooled liquid water looks like the separation of liquid water and vapor just below the conventional critical point.