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The work describes a hybrid algorithm for the dynamic formation of a robot's movement route in nondeterministic environments with bypassing stationary and nonstationary obstacles for two-dimensional space, based on the integration of wave and ant algorithms, which makes it possible to build trajectories of minimum length in real time with simultaneous optimization of a number of criteria for the quality of the constructed path. Restrictions preventing the construction of a trajectory from the current position are identified during the construction process. The trajectory is constructed step by step. The entire trajectory connecting the robot's initial position with the target position is a collection of individual sections. The time complexity of the algorithm depends on the lifespan of the colony, l (number of iterations); the number of graph vertices, n; the number of ants, m; and is estimated as O(ln2m). [ABSTRACT FROM AUTHOR]

Objectives: Distances are functions of spatial positions. Precisely revealing the functional relationship which quantitatively embodies the transmission of uncertainty from spatial positions to their distance, a key scientific problem in need of being solved urgently in geomatics, has important theoretical and practical significance. Methods: Aiming at the limitation of presently available solution of the above mentioned problem, under the premise of that the real position corresponding with the observed one of an uncertain point follows the complete spatial random distribution within the error circle, we have derived the probability distribution function of the distance uncertainty and the corresponding density function containing an uncertain point and those between two uncertain points respectively in two-dimensional space. The latter has been employed to explore the transmission law of point uncertainties to distance uncertainties, opening up a new way for studying and solving the problem of distance uncertainties. Results: The results show that for all cases: (1) When the radius of the error circle (corresponding to the point position accuracy) and the observed distance between points change simultaneously, their ratio has a significant positive correlation with the level of distance uncertainties. (2) When the former remains constant, the distance uncertainty has a significant negative correlation with the latter. (3) When the latter remains constant, the distance uncertainty has a significant positive correlation with the former. Conclusions: As far as the distance uncertainty of cases containing an uncertain point and the one of those between two uncertain points are concerned, the latter is obviously greater than the former when the radius of the error circle and the observed distance between points are consistent for both of them. Otherwise they are not comparable. [ABSTRACT FROM AUTHOR]

Addressing socio-economic development issues are strategic and most important for any country. Multidimensional statistical analysis methods, including comprehensive index assessment, have been successfully used to address this challenge, but they don't cover all aspects of development, leaving some gap in the development of multidimensional metrics. The purpose of the study is to construct a latent metric space based on the use of multidimensional scaling. Based on statistics showing the economic development of Ukrainian regions, two-dimensional space of latent scales was constructed and Ukrainian's regions were positioned in this space. The results were interpreted meaningfully. This use of multidimensional statistical analysis confirms its usefulness for measuring the economic development of regions and allows their comprehensive assessment and comparison.

Grid cells are crucial in path integration and representation of the external world. The spikes of grid cells spatially form clusters called grid fields, which encode important information about allocentric positions. To decode the information, studying the spatial structures of grid fields is a key task for both experimenters and theorists. Experiments reveal that grid fields form hexagonal lattice during planar navigation, and are anisotropic beyond planar navigation. During volumetric navigation, they lose global order but possess local order. How grid cells form different field structures behind these different navigation modes remains an open theoretical question. However, to date, few models connect to the latest discoveries and explain the formation of various grid field structures. To fill in this gap, we propose an interpretive plane-dependent model of three-dimensional (3D) grid cells for representing both two-dimensional (2D) and 3D space. The model first evaluates motion with respect to planes, such as the planes animals stand on and the tangent planes of the motion manifold. Projection of the motion onto the planes leads to anisotropy, and error in the perception of planes degrades grid field regularity. A training-free recurrent neural network (RNN) then maps the processed motion information to grid fields. We verify that our model can generate regular and anisotropic grid fields, as well as grid fields with merely local order; our model is also compatible with mode switching. Furthermore, simulations predict that the degradation of grid field regularity is inversely proportional to the interval between two consecutive perceptions of planes. In conclusion, our model is one of the few pioneers that address grid field structures in a general case. Compared to the other pioneer models, our theory argues that the anisotropy and loss of global order result from the uncertain perception of planes rather than insufficient training. [ABSTRACT FROM AUTHOR]

Grid cells are crucial in path integration and representation of the external world. The spikes of grid cells spatially form clusters called grid fields, which encode important information about allocentric positions. To decode the information, studying the spatial structures of grid fields is a key task for both experimenters and theorists. Experiments reveal that grid fields form hexagonal lattice during planar navigation, and are anisotropic beyond planar navigation. During volumetric navigation, they lose global order but possess local order. How grid cells form different field structures behind these different navigation modes remains an open theoretical question. However, to date, few models connect to the latest discoveries and explain the formation of various grid field structures. To fill in this gap, we propose an interpretive plane-dependent model of three-dimensional (3D) grid cells for representing both two-dimensional (2D) and 3D space. The model first evaluates motion with respect to planes, such as the planes animals stand on and the tangent planes of the motion manifold. Projection of the motion onto the planes leads to anisotropy, and error in the perception of planes degrades grid field regularity. A training-free recurrent neural network (RNN) then maps the processed motion information to grid fields. We verify that our model can generate regular and anisotropic grid fields, as well as grid fields with merely local order; our model is also compatible with mode switching. Furthermore, simulations predict that the degradation of grid field regularity is inversely proportional to the interval between two consecutive perceptions of planes. In conclusion, our model is one of the few pioneers that address grid field structures in a general case. Compared to the other pioneer models, our theory argues that the anisotropy and loss of global order result from the uncertain perception of planes rather than insufficient training.

In this paper, the problem of cooperative control of robotic sensor networks (RSNs) for monitoring dynamic pollutant plumes in two‐dimensional (2D) space is studied. The pollutant plume propagation dynamics is governed by a 2D advection‐diffusion partial differential equation (PDE), and the plume front is modelled by a level set with a pre‐specified threshold value. Distributed consensus Luenberger‐type PDE observers are first constructed using local concentration measurements from the RSNs for unanimous estimate of the dynamic concentration field over the entire spatial domain. With the aid of the distributed consensus observers, a leaderless cooperative control scheme is then developed for the RSNs to monitor the dynamic plume front expansion. It is proved using the Lyapunov stability method and set stability concept that the proposed cooperative control scheme guarantees tracking of the dynamic plume front expansion and coverage of the plume front. Extensive numerical simulation results demonstrate the effectiveness and merit of the proposed cooperative control scheme. [ABSTRACT FROM AUTHOR]

In this study, based on the finite‐time sliding mode control method, an antiswing control law is designed for the underactuated crane systems in two‐dimensional space with external disturbance. The finite‐time disturbance observer is utilised to estimate the external disturbance and develop the finite‐time control law. The designed controller can regulate the trolley to the planned trajectory within a finite time in the presence of external disturbance. Then it can be shown that the proposed control approach can achieve precise trolley positioning and swing suppression. Simulation results are provided to show the satisfactory control performances of the presented control method in terms of working efficiency as well as robustness with respect to external disturbance. [ABSTRACT FROM AUTHOR]

A viscoelastic beam in a two-dimensional space is considered with nonlinear tension. A boundary feedback is applied at the right boundary of the beam to suppress the undesirable vibration. The well-posedness of the problem is established. With the multiplier method, a uniform decay result is proven. [ABSTRACT FROM AUTHOR]

A flexible crane system with vibrating and varying cable is investigated in two‐dimensional space. Two partial differential equations and four ordinary differential equations derived by the Hamilton's principle are used to describe the dynamics of the flexible crane system. The dynamic model of the crane system considers the variation of the tension of the cable. Boundary control design is given to suppress vibrations of the flexible crane system. The Lyapunov's direct method is employed to prove the uniform ultimate boundedness of the states of the cable system. The effectiveness and performance of the proposed control schemes are depicted via numerical simulations. [ABSTRACT FROM AUTHOR]

This paper concerns the Cauchy problem of the two-dimensional full compressible magnetohydrodynamic equations with zero heat-conduction and vacuum as far field density. In particular, the initial density can have compact support. We prove that the Cauchy problem admits a local strong solution provided both the initial density and the initial magnetic field decay not too slow at infinity. [ABSTRACT FROM AUTHOR]

In this study, a new noise model is presented to address the issue of finding an emitting target using time difference of arrival measurements, based on the range between the emitter and a known sensor. To improve the performance of the estimator under the proposed noise model, a weighted version of the spherical interpolation method is proposed and then two weighting matrices required in the method are derived in two different conditions. A detailed theoretical error analysis associated with this algorithm is presented and the Cramer–Rao lower bound is also derived. Simulation studies verify the validity of the proposed error analysis. In addition, in a two‐dimensional space and in the case of a minimal number of sensors, the authors analytically determine the sensors layout in which the location solution is not unique. Via simulations, several placements in a covered region are studied to select the appropriate placement in which the root mean square error of the target position estimation is minimised. Furthermore, simulation results show that they can do this work by the derived expression of the error analysis, which leads to the same outcome. [ABSTRACT FROM AUTHOR]

This work focuses on the construction of rigid formation from non‐rigid ones in the two‐dimensional space. Analogously to operations of Henneberg sequence aiming to guarantee the minimal rigidity of formation, two new operations are introduced, allowing one to sequentially build any rigid graph by connecting non‐rigid ones. A systematic construction sequence is developed based on proposed operations, and is shown to be able to restore rigidity by introducing minimum number of new edges during the construction process. Further applications of the proposed operations are also presented, one of which is successfully employed in the problem of persistence analysis of directed graphs, and can verify the persistence of a given graph with a speed two times faster comparing with existing solution. [ABSTRACT FROM AUTHOR]

• We derived the probability distribution functions and probability density functions on distance uncertainty measurement. • We explore the law of uncertainty transmitting from endpoint positions to distances by means of the established models. • We verified the effectiveness and advantages of our method by comparing it with Monte Carlo method in the case study. Distances are functions of spatial positions, therefore distance uncertainties should be resulted from transmission of spatial position uncertainties via the functional relationships accordingly. How to model the transmission precisely, a challenging problem in GIScience, has increasingly drawn attentions during the past several decades. Aiming at the limitations of presently available solutions to the problem, this article derived probability distribution functions and corresponding density functions of distance uncertainties in two-dimensional space related to one or two uncertain endpoints respectively, under the premise that real positions, corresponding with the observed position of an uncertain point, follow the kernel function within its error circle. The density functions were employed to explore the diffusing law of uncertainty information from point positions to distances, which opened up a new way for thoroughly solving problems of measuring distance uncertainties. It turns out that the proposed methods in this article are more efficient, robust than the corresponding Monte Carlo ones, which verifies their effectiveness and advantages. [ABSTRACT FROM AUTHOR]

It is an important research subject for the fields of cybernetics, robotics and artificial intelligence science that on the basis of neurophysiology and neuroanatomy, simulates and replicates the cerebellar sensorimotor control system for the robots. Around this theme and based on Hoff and Arbib׳s control theory of the minimum jerk, this paper presents a new control model with cerebellar-like structure which can account for the temporal coordination of arm transport and hand preshape during reach and grasp tasks. And it has also been suggested that how the structure could learn the two key functions required in the Hoff–Arbib theory, namely state look-ahead and TTG (time-to-go) estimation. By the simulation for two-dimensional motion of arm transport and hand preshape, the results demonstrate that some key features of human reach–grasp kinematics obtained by Hoff–Arbib model can be achieved by the cerebellum control model and some performances are even better. In a word, by learning and training, the model can obtain the accurate smooth motor trajectory. [ABSTRACT FROM AUTHOR]

This paper concerns the Cauchy problem for nonhomogeneous incompressible Navier–Stokes equations in the two-dimensional (2D) space. By means of the weighted initial density, we obtain the existence and uniqueness of local strong solution with vacuum as a far field state. Moreover, the solution is shown to be globally defined in time, as long as the L ∞ -norm of a weighted density keeps bounded. [ABSTRACT FROM AUTHOR]

This paper concerns the Cauchy problem of the barotropic compressible Navier–Stokes equations on the whole two-dimensional space with vacuum as far field density. In particular, the initial density can have compact support. When the shear and the bulk viscosities are a positive constant and a power function of the density respectively, it is proved that the two-dimensional Cauchy problem of the compressible Navier–Stokes equations admits a unique local strong solution provided the initial density decays not too slow at infinity. Moreover, if the initial data satisfy some additional regularity and compatibility conditions, the strong solution becomes a classical one. [ABSTRACT FROM AUTHOR]

In this study, a robust image watermarking method based on geometric modelling is presented. Eight samples of wavelet approximation coefficients on each image block are utilised to construct two line segments in the two‐dimensional space. The authors change the angle formed between these line segments for data embedding. Geometrical tools are used to solve the tradeoff between the transparency and robustness of the watermark data. They analytically determine the probability density function of the embedding angle for Gaussian samples. Maximum‐likelihood decoder is implemented in the receiver side. Owing to embedding in the angle between two line segments, the proposed scheme has high robustness against the gain attacks. In addition, using the low frequency components of the image blocks for data embedding, high robustness against noise and compression attacks has been achieved. Experimental results confirm the validity of the theoretical analyses given in this study and show the superiority of the proposed method against common attacks, such as Gaussian filtering, median filtering and scaling attacks. [ABSTRACT FROM AUTHOR]

In this study, we consider an acyclic rigid formation with a group of mobile autonomous agents moving in a two‐dimensional space. The formation is generated via a Henneberg sequence construction in which there is one global leader that does not follow any other agents, one first‐follower that only follows the global leader, and each of other agents has two leaders, which is added by a vertex addition or an edge splitting operation. The entire formation moves with the leadership of the global leader. Every follower agent tries to maintain distances towards its leaders. Under the constraint of the acceleration for the global leader, the distributed formation control laws are proposed for the followers that only use the locally relative distance measurement. The control law of the first‐follower is proposed, which needs to know the velocity of the global leader and the relative distance between the global leader and itself. The global asymptotic stability of the expected formation is proved via a Lyapunov‐based technique for the considered multi‐agent system. Moreover, the stable rigidity problem of a formation is investigated for the proposed distributed relative position‐only formation control law. Necessary and sufficient conditions are provided that must be satisfied by the architecture of the underlying graph. Simulation results illustrate the effectiveness of the proposed formation control approach. [ABSTRACT FROM AUTHOR]

The problem of existence of bounded relativistic fermion states in a spherically symmetric well of finite depth in a two-dimensional space is investigated. The well depth critical for the appearance of standard states with energies E = m, 0, and - m is determined; moreover, cases with zero and nonzero fermion momenta are considered. Approximate analytical expressions for the critical depths of narrow and wide wells are derived which are in good agreement with the results of numerical calculations. Approximate energies of levels located on the boundaries of the upper and lower continuums and determined analytically are in good agreement with the results of numerical calculations. [ABSTRACT FROM AUTHOR]

The main phenomenon of coagulation is that particles aggregate to form floc. The formation of floc and cluster was simulated with computer in two-dimensional space using particle-cluster aggregation model and the properties of floc were analyzed using fractal theory. The effect of sticking probabilities on floc structure, density and porosity were analyzed. With the sticking probability increases, the simulated floc structure is gradually loose, floc density is gradually lower, the porosity is greater and the fractal dimensions are decrease, which are not conducive to precipitation. Through this study the factors that can not be accurately studied in the previous experiments could be controlled, which provided an effective new way for the research of particle aggregation. [ABSTRACT FROM AUTHOR]

In the scaling of emotions in general, and their application to music in particular, the valence (good/bad) and activity or arousal dimensions are ubiquitous. Naive intuition and critics' writings have assumed a greater profundity in music's ability to express emotions than this would imply. Five experiments were performed to show that music is capable of expressing a greater detail in emotional range than can be captured by these two dimensions. The basic paradigm had participants rank order sets of emotions with respect to how well they applied to brief, unfamiliar instrumental selections. People did so rapidly, with significant reliability, even when the affective compass of the emotions to be ranked was restricted to enforce attention to subtle withindimension distinctions, or expanded to allow attention to subtle distinctions. [ABSTRACT FROM PUBLISHER]

By using a new Hamiltonian of interaction we have calculated the interaction energy for two-dimensional and three-dimensional lattices. We present also, approximate analytical formulae and the analytical formulae for the constant of the elastic force. The obtained results show that in the three-dimensional space, the two-dimensional lattice has the lattice constant and the cohesive energy which are smaller than that of the three-dimensional lattice. For appropriate values of the coupling constants, the two-dimensional lattice in a two-dimensional space has both the lattice constant and the cohesive energy, larger than that of the two-dimensional lattice in a three-dimensional space; this means that if there is a two-dimensional space in the Universe, this should be thinner than the three-dimensional space, while the interaction forces should be stronger. On the other hand, if the coupling constant in the two-dimensional lattice in the two-dimensional space is close to zero, the cohesive energy should be comparable with the cohesive energy from three-dimensional space but this two-dimensional space does not emit but absorbs radiation. [ABSTRACT FROM AUTHOR]