Convergence Of Trajectories Research Articles

On the Application of Good-Turing Statistics to Quantify Convergence of Biomolecular Simulations

Quantifying convergence and sufficient sampling of macromolecular molecular dynamics simulations is more often than not a source of controversy (and of various ad hoc solutions) in the field. Clearly, the only reasonable, consistent, and satisfying way to infer convergence (or otherwise) of a molecular dynamics trajectory must be based on probability theory. Ideally, the question we would wish to answer is the following: "What is the probability that a molecular configuration important for the analysis in hand has not yet been observed ?". Here we propose a method for answering a variant of this question by using the Good-Turing formalism for frequency estimation of unobserved species in a sample. Although several approaches may be followed in order to deal with the problem of discretizing the configurational space, for this work we use the classical RMSD matrix as a means to answering the following question: "What is the probability that a molecular configuration with an RMSD (from all other already observed configurations) higher than a given threshold has not actually been observed ?". We apply the proposed method to several different trajectories and show that the procedure appears to be both computationally stable and internally consistent. A free, open-source program implementing these ideas is immediately available for download via public repositories.

Breaking the Glass Ceiling? The Effect of Board Quotas on Female Labor Market Outcomes in Norway

In late 2003, Norway passed a law mandating 40 percent representation of each gender on the board of publicly limited liability companies. The primary objective of this reform was to increase the representation of women in top positions in the corporate sector and decrease gender disparity in earnings within that sector. We document that the newly (post-reform) appointed female board members were observably more qualified than their female predecessors, and that the gender gap in earnings within boards fell substantially. While the reform may have improved the representation of female employees at the very top of the earnings distribution (top 5 highest earners) within firms that were mandated to increase female participation on their board, there is no evidence that these gains at the very top trickled-down. Moreover the reform had no obvious impact on highly qualified women whose qualifications mirror those of board members but who were not appointed to boards. We observe no statistically significant change in the gender wage gaps or in female representation in top positions, although standard errors are large enough that we cannot rule economically meaningful gains. Finally, there is little evidence that the reform affected the decisions of women more generally; it was not accompanied by any change in female enrollment in business education programs, or a convergence in earnings trajectories between recent male and female graduates of such programs. While young women preparing for a career in business report being aware of the reform and expect their earnings and promotion chances to benefit from it, the reform did not affect their fertility and marital plans. Overall, in the short run the reform had very little discernible impact on women in business beyond its direct effect on the newly appointed female board members.

An Improved Path-Generating Regulator for Two-Wheeled Robots to Track the Circle/Arc Passage

The improved path-generating regulator (PGR) is proposed to path track the circle/arc passage for two-wheeled robots. The PGR, which is a control method for robots so as to orient its heading toward the tangential direction of one of the curves belonging to the family of path functions, is applied to navigation problem originally. Driving environments for robots are usually roads, streets, paths, passages, and ridges. These tracks can be seen as they consist of straight lines and arcs. In the case of small interval, arc can be regarded as straight line approximately; therefore we extended the PGR to drive the robot move along circle/arc passage based on the theory that PGR to track the straight passage. In addition, the adjustable look-ahead method is proposed to improve the robot trajectory convergence property to the target circle/arc. The effectiveness is proved through MATLAB simulations on both the comparisons with the PGR and the improved PGR with adjustable look-ahead method. The results of numerical simulations show that the adjustable look-ahead method has better convergence property and stronger capacity of resisting disturbance.

Control of non-linear non-integer-order systems using variable structure control theory

This paper deals with the problem of robust control of uncertain non-integer-order non-linear complex systems in finite time. First, a novel fractional-order integral-type switching manifold is introduced. Then, the fractional-order stability theory is used to prove the finite time stability of the resulting sliding mode dynamics in a given finite time. Afterward, using the variable structure control theory, a novel robust non-integer control law is proposed to guarantee the convergence of the system trajectories to the prescribed sliding manifold within a given finite time. Simulation results reveal that the proposed fractional variable structure controller works well for finite-time chaos suppression of fractional-order hyperchaotic Chen and chaotic Lorenz systems with system uncertainties.

Global asymptotic tracking of asymmetrical underactuated surface vessels with parameter uncertainties

In this work, we investigate the tracking control problem of asymmetrical underactuated surface vessels with parameter uncertainties. The tracking error model is first derived via appropriate coordinate transformations, and is considered as a cascade structure composed of two subsystems. The Lyapunov redesign approach is employed to construct the control laws separately to stabilize the two subsystems with unknown model parameters. The cascade system theory is applied to prove the global uniform asymptotic convergence of the state trajectory to the reference one provided the desired yaw velocity is not vanishing. The effectiveness of the proposed control laws is verified by simulation examples.

Jetlike tunneling from a trapped vortex

We analyze the tunneling of vortex states from elliptically shaped traps. Using the hydrodynamic representation of the Gross-Pitaevskii (Nonlinear Schr\"odinger) equation, we derive analytically and demonstrate numerically a novel type of quantum fluid flow: a jet-like singularity formed by the interaction between the vortex and the nonhomogenous field. For strongly elongated traps, the ellipticity overwhelms the circular rotation, resulting in the ejection of field in narrow, well-defined directions. These jets can also be understood as a formation of caustics since they correspond to a convergence of trajectories starting from the top of the potential barrier and meeting at a certain point on the exit line. They will appear in any coherent wave system with angular momentum and non-circular symmetry, such as superfluids, Bose-Einstein condensates, and light.

Global Asymptotic Trajectory Tracking and Point Stabilization of Asymmetric Underactuated Ships with Non-Diagonal Inertia/Damping Matrices

In this work, we investigate the trajectory tracking and point stabilization problems of asymmetric underactuated surface ships with non-diagonal inertia and damping matrices. By combining the novel state and input transformations, the direct Lyapunov approach, and the nonlinear time-varying tools, the trajectory tracking controller is derived, guaranteeing global κ-exponential convergence of state trajectory to the reference one satisfying mild persistent exciting conditions. By properly designing the reference trajectory, the proposed tracking scheme is also generalized to achieve global uniform asymptotic point stabilization. Simulation examples are given to illustrate the effectiveness of the proposed control schemes.

Biological legacies: Direct early ecosystem recovery and food web reorganization after a volcanic eruption in Alaska

Attempts to understand how communities assemble following a disturbance are challenged by the difficulty of determining the relative importance of stochastic and deterministic processes. Biological legacies, which result from organisms that survive a disturbance, can favour deterministic processes in community assembly and improve predictions of successional trajectories. Recently disturbed ecosystems are often so rapidly colonized by propagules that the role of biological legacies is obscured. We studied biological legacies on a remote volcanic island in Alaska following a devastating eruption where the role of colonization from adjacent communities was minimized. The role of biological legacies in the near shore environment was not clear, because although some kelp survived, they were presumably overwhelmed by the many vagile propagules in a marine environment. The legacy concept was most applicable to terrestrial invertebrates and plants that survived in remnants of buried soil that were exposed by post-eruption erosion. If the legacy concept is extended to include ex situ survival by transient organisms, then it was also applicable to the island's thousands of seabirds, because the seabirds survived the eruption by leaving the island and have begun to return and rebuild their nests as local conditions improve. Our multi-trophic examination of biological legacies in a successional context suggests that the relative importance of biological legacies varies with the degree of destruction, the availability of colonizing propagules, the spatial and temporal scales under consideration, and species interactions. Understanding the role of biological legacies in community assembly following disturbances can help elucidate the relative importance of colonists versus survivors, the role of priority effects among the colonists, convergence versus divergence of successional trajectories, the influence of spatial heterogeneity, and the role of island biogeographical concepts.

Performance of a class of multi-robot deploy and search strategies based on centroidal voronoi configurations

This article considers a class of deploy and search strategies for multi-robot systems and evaluates their performance. The application framework used is deployment of a system of autonomous mobile robots equipped with required sensors in a search space to gather information. The lack of information about the search space is modelled as an uncertainty density distribution. The agents are deployed to maximise single-step search effectiveness. The centroidal Voronoi configuration, which achieves a locally optimal deployment, forms the basis for sequential deploy and search (SDS) and combined deploy and search (CDS) strategies. Completeness results are provided for both search strategies. The deployment strategy is analysed in the presence of constraints on robot speed and limit on sensor range for the convergence of trajectories with corresponding control laws responsible for the motion of robots. SDS and CDS strategies are compared with standard greedy and random search strategies on the basis of time taken to achieve reduction in the uncertainty density below a desired level. The simulation experiments reveal several important issues related to the dependence of the relative performances of the search strategies on parameters such as the number of robots, speed of robots and their sensor range limits.

Ontrol of Spacecraft Formation with Disturbance Rejection and Exponential Gains

We address the problem of state feedback translational motion control of a spacecraft formation through a modied sliding surface controller using variable gains and I 2 action for disturbance rejection. The exponential varying gains ensure faster convergence of the state trajectories during attitude maneuver while keeping the gains small (and the system less sti) for station keeping. Integral action is introduced for rejection of disturbances with a constant nonzero mean such as aerodynamic drag. A direct consequence is a drop in energy consumption when aected by sensor noise and a decrease in size of the error states residual when operating close to the equilibrium point. A large number of simulation results are presented to show the control performance.

Delayed Feedback Control for Linear Time-Varying Periodic Systems in Act-and-Wait Fashion

A delayed feedback control framework for stabilizing periodic orbits of a linear periodic time-varying system is proposed. Act-and-wait approach is utilized for switching the proposed delayed feedback controller on and off alternately at every integer multiples of the period of the system. We obtain conditions for the convergence of the state trajectories towards a periodic solution of the closed-loop system under our proposed control law. We present a numerical example to illustrate the efficacy of our approach.

Suppression of the wing-rock phenomenon using nonlinear controllers

This paper deals with the design of nonlinear controllers for the wing-rock phenomenon of a delta wing aircraft. A fifth order dynamic model is used to describe this phenomenon. A state transformation is introduced such that the transformed dynamic model is in a form which is suitable for a variety of control designs. A feedback linearization control scheme and a sliding-mode control (SMC) scheme are then proposed to suppress wing rock oscillations. It is shown that the two controllers successfully suppress the undesired oscillations and guarantee the asymptotic convergence of all system trajectories to their desired values. The effectiveness of the proposed controllers is verified through simulation studies.

Kinematic and Dynamic Analysis of North-Tehran Tectonic Wedge Formed in South Central Alborz, Iran

Abstract To answer the question “Whether the North-Tehran tectonic wedge is a dynamic tectonic wedge or not?” we applied paleostress techniques to investigate fault slip data. The mean reduced stress tensor is defined for all stabilized stress regions. Unscaled Mohr’s circles drawn for fault slip data were used to obtain the relative slip tendency of clusters on diagrams. It showed that the slip tendency in the vicinity of fault junction is much lower than expected. The mean σ1 defined for a combination of fault slip data trends N14°E nearly parallel with the overall pressure in Iranian crust at the latitude of Central Alborz. This trend suggests the least effect of boundary faults and the wedge between them on stress orientations inside the wedge. Finally, the stress trajectory map was prepared showing the configuration and relative intensity of σ1. The map did not illustrate any convergence in σ1 trajectories and the consequent concentration of stress and seismic potential in fault junction. That is a direct evidence for disagreeing the dynamicity of this tectonic wedge.

On a double penalized Smectic-A model

In smectic-A liquid crystals, a unity director vector $\boldsymbol{n}$ appear modeling an average preferential direction of the molecules and also the normal vector of the layer configuration. In the E's model [5], the Ginzburg-Landau penalization related to the constraint $|\boldsymbol{n}|=1$ is considered and, assuming the constraint $\nabla\times \boldsymbol{n}=0$, $\boldsymbol{n}$ is replaced by the so-called layer variable $\varphi$ such that $\boldsymbol{n}=\nabla\varphi$.  &nbsp In this paper, a double penalized problem is introduced related to a smectic-A liquid crystal flows, considering a Cahn-Hilliard system to model the behavior of $\boldsymbol{n}$. Then, the issue of the global in time behavior of solutions is attacked, including the proof of the convergence of the whole trajectory towards a unique equilibrium state.

Nonlinear model predictive control from data: a set membership approach

SUMMARYA new approach to design a Nonlinear Model Predictive Control law that employs an approximate model, derived directly from data, is introduced. The main advantage of using such models lies in the possibility to obtain a finite computable bound on the worst‐case model error. Such a bound can be exploited to analyze the robust convergence of the system trajectories to a neighborhood of the origin. The effectiveness of the proposed approach, named Set Membership Predictive Control, is shown in a vehicle lateral stability control problem, through numerical simulations of harsh maneuvers. Copyright © 2012 John Wiley & Sons, Ltd.

Stability of interconnected impulsive switched systems subject to state dimension variation

The contribution of this article is twofold. First we deal with the stability of continuous-time interconnected impulsive switched systems for which the dimension of the system’s state vector may change at different modes. A dwell time condition is derived to ensure the exponential convergence of the state trajectories, and when the system is subject to a non-vanishing perturbation, a superior bound of the state trajectories is provided as well. The main advantage of the solution we propose is its direct application to practical fields such as metallurgy or traffic control through the use of semi-definite programming solvers. Second, we introduce in the framework a platoon of vehicles in automated highway systems where vehicles may join or leave the platoon. Then, we detail the longitudinal stability problem of such a system and illustrate the features of the proposed stability conditions through numerical simulations.

Compensatory growth after various levels of dietary protein restriction in rainbow trout, Oncorhynchus mykiss

This study was conducted to determine the effects of various levels of dietary protein restriction for 3weeks (phase 1) on compensatory growth, nutrient utilization, body compositions and organ indices of rainbow trout (Oncorhynchus mykiss) during a nine-week realimentation phase (phase 2). Five experimental treatments consisted of four protein restricted diets including 15, 22.5, 30 and 37.5% protein (RP15, RP22.5, RP30 and RP37.5 respectively) and the control diet 45% (C45). All diets were prepared to be isoenergetic at levels of 22.5MJkg−1 gross or 16.3MJkg−1 digestible energy. Each of five treatment groups was tried in triplicate groups of 75 fish with an average initial weight of 62.06±0.38g. Protein restriction in phase 1 linearly decreased specific growth rates of weight and length (SGRW and SGRL), body weight and length as well as feed utilization. The restriction induced a linearly increasing trend in protein retention efficiency, feed intake, organ indices such as viscera, pyloric caecum and hepato-somatic, whole body lipid and lipid/lean body mass whereas it induced a decrease in whole body moisture and protein at the end of phase 1. Protein restriction, except RP37, caused a significant increase in SGRW during phase 2 compared with C45, being most pronounced in the first 3weeks. Repeated measures ANOVA revealed a convergence in trajectories of ln body weight and length at the end of the experiment, suggesting that a complete catch up occurred in body mass and structure. There was also replenishment in distorted organ indices and body compositions, except protein which was still less in R15 than C45. More efficient utilization of feed and dietary protein by previously protein restricted fish during phase 2 appears to be the reason of the compensatory growth. At the end of the experiment, protein restriction was found to be effective in reduction of N loss.

Modeling and solving continuous-time instantaneous dynamic user equilibria: A differential complementarity systems approach

This paper is the second of a two-part research wherein we undertake a mathematically rigorous investigation of the continuous-time dynamic user equilibrium (DUE) problem using the recently introduced mathematical paradigm of differential complementarity systems (DCSs). Based on the thorough study of continuous-time single-destination point-queue models in the previous part, we first extend this special case to multiple destinations respecting the First-In–First-Out property of travel flows. A DCS with constant time delay is then introduced to formulate the continuous-time model of instantaneous dynamic traffic equilibria (IDUE) with a fixed demand profile. We develop a time decomposition scheme based on link free flow travel times to convert the delay DCS to a series of DCSs without time delays that are solved by a numerical time-stepping method. We provide rigorous numerical treatment of the time-decomposed IDUE model, including solvability of the discrete-time complementarity problems and convergence of the numerical trajectories to a continuous-time solution. We present numerical results to validate the IDUE on a small network and also on the Sioux Falls network.

Continuous-time point-queue models in dynamic network loading

Extending a recent work (Pang et al., in press) pertaining to a simple single-bottleneck model, this paper is the first of a two-part research wherein we undertake a mathematically rigorous study of the continuous-time dynamic user equilibrium (DUE) problem using the recently introduced mathematical paradigm of differential complementarity systems (DCSs) (Pang and Stewart, 2008). The first step in this comprehensive research is to gain a thorough understanding of some continuous-time point-queue models, which will be used as the building block of a computationally tractable model for the continuous-time DUE problem that we will study in detail in the accompanying paper (Ban et al., in press). Starting with the original point-queue model introduced by Vickrey (1969), we summarize some desirable properties that a continuous-time point-queue model should possess, and show that one of these properties—the nonnegativity of the queue lengths—is violated by Vickrey’s original model. As a remedy to this drawback of Vickrey’s model and with the goal of extending it to a continuous-time setting, we introduce two continuous-time point-queue models and show that they satisfy the properties we propose. Discretizations of the continuous-time models are discussed and construction of numerical trajectories is presented; convergence of such trajectories as the time step approaches zero is established; regularity of a solution to the continuous-time problem is clarified, and numerical results are presented.

ON DYNAMICS OF ξS QUADRATIC STOCHASTIC OPERATORS

In this research we introduce a new class of quadratic stochastic operators called ξ s -QSO which are defined through coefficient of the operator from measure-theoretic (namely we are looking the coefficient as the measures which are absolute continuous or singular) point of view. We also study the limiting behaviour of ξ s -QSO defined on 2D-simplex. We first describe ξ s -QSO on 2D-simplex and classify them with respect to the conjugacy and renumeration of the coordinates. We find six non-isomorphic classes of such operators. Moreover, we investigate the behaviour of each operator from three classes and prove convergence of trajectories of these classes and study their certain properties. We showed trajectories of two classes converge to the equilibrium. For the third class, it is established only the negative trajectories converge to the equilibrium.