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We study binary word operations of the insertion, deletion and substitution type. Many of these operations can be generalized into a unified framework by introducing so-called trajectory condition. This generalization has been previously made for insertion and deletion operations. In this paper we naturally extend this approach also to substitution operations. We study closure properties and decision problems of substitutions on trajectories. The obtained results are then applied to model complex noisy channels and a cryptanalysis problem. Another application concerns the design of sets of DNA strands without undesired bonds.

We examine and model dynamics in three areas of social cognition: (1) political transformations within Russia, (2) evaluation of political trends in other countries by Russians, and (3) evaluation of Russian stereotypes concerning women. We try to represent consciousness as vectorfields and trajectories in a cognitive state space. We use psychosemantic techniques that allow definition of the state space and the systematic construction of these vectorfields and trajectories and their portrait from research data. Then we construct models to fit them, using multiple regression methods to obtain linear differential equations. These dynamical models of social cognition fit the data quite well. (1) The political transformations were modeled by a spiral repellor in a two-dimensional space of a democratic–totalitarian factor and social depression–optimism factor. (2) The evaluation of alien political trends included a flow away from a saddle toward more stable and moderate political regimes in a 2D space, of democratic–totalitarian and unstable–stable cognitive dimensions. (3) The gender study showed expectations (attractors) for more liberated, emancipated roles for women in the future.

The topological solutions of the De Broglie–Bohm quantum mechanics are presented. Starting from the Schrödinger equation for one particle system and ϕ-mapping topological current theory, the trajectory of the particle is derived explicitly, and can be used as the world line of the particle. The world line is just at the zero point of the wave function and it is shown that the vorticity of the world line can be expressed by Hopf index and Brouwer degree. The evolution of the world line at the bifurcation point is given.

The free falling motion of round thin disks in quiescent water was investigated experimentally. The mean density of the disks were about 1.05 g/cm³, slightly larger than water, and thickness to diameter ratio is 0.1. In this letter, We reported two cases with different mass distributions, in one case the disk's center of mass coincided with the geometry center and in another case it was deviated. Two CCD cameras were used to photograph the falling disk simultaneously, the disk's six degrees of freedom were obtained via a stereoscopic vision method. In the first case fluttering motion was observed, the disk's oscillatory motion confined in a vertical plane, and the pitching motion occurred about an axis normal to the plane. In the second case, steady helical falling was observed, the disk moved in a helical path at constant speed, the attacking angle is constant.

To address the problem of low accuracy of aircraft flight trajectory, a new prediction method for flight trajectory is proposed based on the quasi-Newton and mesh simplification algorithm. This method firstly proposes the calculation method of 3D aircraft flight trajectory, and simplifies the calculation process with mesh simplification algorithm. At the same time, the prediction accuracy is further improved with the quasi-Newton algorithm. At last, through the simulation experiment, we have made a deep study of the key factors influencing the prediction method. The result has shown that the method has good adaptability.

In order to meet the motion characteristics requirements of multi-objective fusion of execution components, a synthesis method of open-chain coupled serial linkage mechanism fusing trajectory and posture is proposed. According to periodic change characteristics of the execution component trajectory and posture of the linkage mechanism, the motion characteristics equation including trajectory and posture information is established by using Fourier series. The harmonic parameters of the link trajectory and posture are calculated according to the principle of discrete Fourier transform. The complex vector theory is applied to determine the functional relationship between the harmonic parameters of the trajectory and posture and the independent variable of the linkage mechanism, and furthermore the motion synthesis equation of the open-chain linkage mechanism fusing trajectory and posture information is established. The linkage mechanism parameters are calculated using Mathcad Prime software. The single-degree-of-freedom open-chain coupled serial linkage mechanism fusing trajectory and posture information is obtained by using a noncircular gear planetary gear train to couple the connection points of the linkage mechanism. The correctness of the method is verified through an example.

In this paper, the complex dynamical behaviors of the chaotic trajectories of Chen's system are analyzed in detail, with its precise bound derived for the first time. In particular, it is rigorously proved that all nontrivial trajectories of the system always travel alternatively through two specific Poincaré projections for infinitely many times. The results provide an insightful understanding of the complex topological structure of Chen's chaotic attractor.

This paper analyzes the convergence properties and convergence region of a class of trajectory-based power flow methods. The convergence region of the trajectory-based method is a connected set and possesses the near-by property. The convergence region of Newton method and trajectory-based method in solving power flow problems are numerically investigated. Since the convergence region of trajectory-based method corresponds to the stability region of the nonlinear dynamic system, the stability regions of two dynamic systems are computed. The numerical results indicate that the stability region of the dynamic system possesses better geometry features than the convergence region of Newton method. These properties make the trajectory-based power flow method robust, especially on heavy loading conditions.

One of the fascinating aspects of sports rivalry is that anything can happen. The significant difficulty is that computer-aided systems must address how to record and analyze many game events, and fractal AI plays an essential role in dealing with complex structures, allowing effective solutions. In table tennis, we primarily concentrate on two issues: ball tracking and trajectory prediction. Based on these two components, we can get ball parameters such as velocity and spin, perform data analysis, and even create a ping-pong robot application based on fractals. However, most existing systems rely on a traditional method based on physical analysis and a non-machine learning tracking algorithm, which can be complex and inflexible. As mentioned earlier, to overcome the problem, we proposed an automatic table tennis-aided system based on fractal AI that allows solving complex issues and high structural complexity of object tracking and trajectory prediction. For object tracking, our proposed algorithm is based on structured output Convolutional Neural Network (CNN) based on deep learning approaches and a trajectory prediction model based on Long Short-Term Memory (LSTM) and Mixture Density Networks (MDN). These models are intuitive and straightforward and can be optimized by training iteratively on a large amount of data. Moreover, we construct a table tennis auxiliary system based on these models currently in practice.

The common argument for the small business is that there simply is no strategy; that small businesses react heuristically to events, guided by the whims and passions of the owner-manager. Strategy, however, can be looked at from a behavioural perspective, as opposed to the more normative strategic planning schools that small firms rarely abide by. This concept of strategy as 'behaviour' encompasses the actions of the owner-manager, the context of the small firm, and the consequences of the actions taken. It looks at strategy as part deliberate and part emergent, allowing for the inclusion of both external influences and internal decision making. Grounded theory research on small firms in Malta has in fact shown strategy to be a dynamic phenomenon, one that can be viewed as a set of defined pathways between identifiable life cycle states. The paper shall outline the research findings that have identified five distinct patterns of small firm strategic behaviour, each with its own unique trajectory and performance implications. Understanding which strategic pathway a small firm belongs to allows for a comprehensive insight into the firm's competitive behaviour, and a prediction of the consequences of that behaviour.

The history of the quadratic stochastic operators can be traced back to the work of Bernshtein (1924). For more than 80 years, this theory has been developed and many papers were published. In recent years it has again become of interest in connection with its numerous applications in many branches of mathematics, biology and physics. But most results of the theory were published in non-English journals, full text of which are not accessible. In this paper we give all necessary definitions and a brief description of the results for three cases: (i) discrete-time dynamical systems generated by quadratic stochastic operators; (ii) continuous-time stochastic processes generated by quadratic operators; (iii) quantum quadratic stochastic operators and processes. Moreover, we discuss several open problems.

In archery, there is the relationship among stability of the archer, adjustments made during the aiming procedure, and the targeting coordination. Twelve specialists from the male archery team of National College of Physical Education and Sports took part in this research, with 12 arrows sets up at a distance of 70 m. A laser pen is located on the bow handle for capturing the trajectory of the aiming procedure with a digital video camera, and the other camera for the coordination of target. The Ariel Performance Analysis System motion analysis system is used to obtain the x and y aiming trajectory coordinates for analyzing the correlation. Results of this research were: (1) Ten archers whose variation of the x axis is larger than that of the y axis. (2) There are four archers whose aiming adjustment procedure has all the positive correlation with the target in the x axis, but there are three archers with all negative correlation. (3) In terms of the correlation in the y axis, there are six archers with all positive correlation, but there are two archers with all negative correlation.

We point to the connection between a recently introduced class of non-Markovian master equations and the general structure of quantum collisional models. The basic construction relies on three basic ingredients: a collection of time dependent completely positive maps, a completely positive trace preserving transformation and a waiting time distribution characterizing a renewal process. The relationship between this construction and a Lindblad dynamics is clarified by expressing the solution of a Lindblad master equation in terms of demixtures over different stochastic trajectories for the statistical operator weighted by suitable probabilities on the trajectory space.

This paper describes in detail the control architecture of an anthropometric robotic leg. It contains a brief description of the mechanical structure of the prototype and a kinematical structure overview. The aim of this project is the development of new technologies for the implementation and realization of bio-mechanical limbs for motoric gait rehabilitation, as well as for human gate study and analysis. In particular, a detailed description of the pneumatic control architecture is provided, together with some design considerations. Moreover, a platform for walking trajectory generation is presented, and some experimental results are reported. In order to assure a proper stability control during the gait, a center of pressure sensor has been specifically designed and placed onboard the robot. A detailed description on the sensor mechanism is also provided.

This paper presents a new third-order trajectory solution in Lagrangian form for the water particles in a wave-current interaction flow based on an Euler–Lagrange transformation. The explicit parametric solution highlights the trajectory of a water particle and the wave kinematics above the mean water level and within a vertical water column, which were calculated previously by an approximation method using Eulerian approach. Mass transport associated with a particle displacement can now be obtained directly in Lagrangian form. The angular frequency and Lagrangian mean level of the particle motion in Lagrangian form differ from those of the Eulerian. The variations in the wave profile and the water particle orbits resulting from the interaction with a steady uniform current of different magnitudes are also investigated. Comparison on the wave profiles given by the Eulerian and Lagrangian solution to a third-order reveals that the latter is more accurate than the former in describing the shape of the wave profile. Moreover, the influence of a following current is found to increase the relative horizontal distance traveled by a water particle, while the converse is true in the case of an opposing current.

A stay region of a moving object is a region in which it spends a significant amount of time. Given the location of an object through a certain time interval, i.e. its trajectory, our goal is to find its stay regions. We consider axis-parallel square stay regions of fixed side length, which contain the longest continuous portion of the trajectory. We present an algorithm for answering time-windowed stay region queries, to find a stay region in any given time interval. The algorithm has an approximation factor of $1∕2$ and answers each query with the time complexity $O(logn)$, in which $n$ is the number of vertices of the input trajectory. The time complexity of the preprocessing step of the algorithm is $O(n)$. We also present two approximation algorithms for finding stay regions of whole trajectories.

We introduce a notion of ℓ-Volterra quadratic stochastic operator defined on (m - 1)-dimensional simplex, where ℓ ∈ {0,1,…, m}. The ℓ-Volterra operator is a Volterra operator if and only if ℓ = m. We study structure of the set of all ℓ-Volterra operators and describe their several fixed and periodic points. For m = 2 and 3, we describe behavior of trajectories of (m - 1)-Volterra operators. The paper also contains many remarks with comparisons of ℓ-Volterra operators and Volterra ones.

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.

Understanding how proteins fold is essential to our quest in discovering how life works at the molecular level. Current computation power enables researchers to produce a huge amount of folding simulation data. Hence there is a pressing need to be able to interpret and identify novel folding features from them. In this paper, we model each folding trajectory as a multi-dimensional curve. We then develop an effective multiple curve comparison (MCC) algorithm, called the enhanced partial order (EPO) algorithm, to extract features from a set of diverse folding trajectories, including both successful and unsuccessful simulation runs. Our EPO algorithm addresses several new challenges presented by comparing high dimensional curves coming from folding trajectories. A detailed case study of applying our algorithm to a miniprotein Trp-cage²⁴ demonstrates that our algorithm can detect similarities at rather low level, and extract biologically meaningful folding events.

The Lagrangian points for the Sun-Earth system are considered due to their special importance for the scientific community for the design of space missions. The location of the Lagrangian points with the trajectories and stability regions are computed numerically for the initial conditions very close to the points. The influence of belt, effect of radiation pressure due to the Sun and oblateness effect of second primary(finite body Earth) is presented for various values of parameters. The collinear points are asymptotically stable within a specific interval of time t correspond to the values of parameters and initial conditions.