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Ant colony algorithm has a wide range of application value in robot path planning. When the robot encounters obstacles, it can find the optimal path in the model space according to relevant performance indexes, and the collision times from the starting point to the target position are the least. In this paper, the research significance of ant colony algorithm in game map path finding is briefly discussed, and the application of ant colony algorithm in map path finding is explained for readers’ reference.

According to the principle of conservation of mass and the fractional Fick’s law, a new two-sided space-fractional diffusion equation was obtained. In this paper, we present two accurate and efficient numerical methods to solve this equation. First we discuss the alternating-direction finite difference method with an implicit Euler method (ADI–implicit Euler method) to obtain an unconditionally stable first-order accurate finite difference method. Second, the other numerical method combines the ADI with a Crank–Nicolson method (ADI–CN method) and a Richardson extrapolation to obtain an unconditionally stable second-order accurate finite difference method. Finally, numerical solutions of two examples demonstrate the effectiveness of the theoretical analysis.

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.

A key problem in ecology is to predict the presence/absence of species across a geographical region. Addressing this problem in a multi-species community with priority effects (i.e., initial abundances determine the presence/absence of species) is a challenging task because species presence/absence depends on many factors such as abiotic environments, biotic interactions (i.e., interactions among species) and dispersal process. While various ecological factors have been considered, less attention has been given to the problem of understanding how dimensionality of space, in interaction with other factors, shape community assembly in the presence of priority effects. In this paper, we employ partial differential equations models (in one- and two-dimensional space) to examine the consequences of different dimensionality of space on the occurrence of priority effects and species coexistence in multi-species communities. We discover that as well as a pronounced increase in system complexity, adding a second space dimension essentially modifies the strength of priority effect. It is also observed that more outcomes with multi-species coexistence emerge as dimensionality of space is increased. As such, incorporating dimensionality in an ecological system with heterogeneous environments could engender additional insights on species coexistence mechanisms (as compared to just adding an extra dimension in the case of purely homogeneous space). These effects will strongly depend on how each interacting species responds to the environmental gradients. Overall, this study is among the first to explicitly show that combinations of distinct abiotic factors can shape the distributions of multiple competing species that are different from those in single abiotic factor case, and some dispersing species may respond to complex sets of abiotic and biotic conditions in variable ways.

The recent exceptional demand for emotion recognition systems in clinical and non-medical applications has attracted the attention of many researchers. Since the brain is the primary object of understanding emotions and responding to them, electroencephalogram (EEG) signal analysis is one of the most popular approaches in affect classification. Previously, different approaches have been presented to benefit from brain connectivity information. We envisioned analyzing the interactions between brain electrodes with the information potential and providing a new index to quantify the connectivity matrix. The current study proposed a simple measure based on the cross-information potential between pairs of EEG electrodes to characterize emotions. This measure was tested for different EEG frequency bands to realize which EEG waves could be fruitful in recognizing emotions. Support vector machine and k-nearest neighbor (kNN) were implemented to classify four emotion categories based on two-dimensional valence and arousal space. Experimental results on the Database for Emotion Analysis using Physiological signals revealed a maximum accuracy of 90.14%, a sensitivity of 89.71%, and an F-score of 94.57% using kNN. The gamma frequency band obtained the highest recognition rates. Furthermore, low valence-low arousal was classified more effectively than other classes.

This paper develops the approximate solution of the two-dimensional space-time fractional diffusion equation. Firstly, the time-fractional derivative is discretized with a scheme of order O ( δ τ 2 - α ) , 0 < α < 1 . Then, the Chebyshev spectral collocation of the third kind is implemented to approximate spatial variables and to obtain full discretization of the equation. Moreover, the unconditional stability and convergence of the proposed method are shown in the perspective H 2 -norm. Two numerical examples are presented to verify the effectiveness and the accuracy of the proposed method. The comparison between our obtained numerical results and the results of existing schemes in the literature shows that the proposed method is more reliable and precise.

This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the question turns out to be closely related to porous sets in Banach spaces, it provides a bridge between descriptive set theory and the classical topic of existence of derivatives of vector-valued Lipschitz functions. The topic is relevant to classical analysis and descriptive set theory on Banach spaces. The book opens several new research directions in this area of geometric nonlinear functional analysis. The new methods developed here include a game approach to perturbational variational principles that is of independent interest. Detailed explanation of the underlying ideas and motivation behind the proofs of the new results on Fréchet differentiability of vector-valued functions should make these arguments accessible to a wider audience. The most important special case of the differentiability results, that Lipschitz mappings from a Hilbert space into the plane have points of Fréchet differentiability, is given its own chapter with a proof that is independent of much of the work done to prove more general results. The book raises several open questions concerning its two main topics.

Most existing research on applying the finite element method to discretize space fractional operators is studied on regular domains using either uniform structured triangular meshes, or quadrilateral meshes. Since many practical problems involve irregular convex domains, such as the human brain or heart, which are difficult to partition well with a structured mesh, the existing finite element method using the structured mesh is less efficient. Research on the finite element method using a completely unstructured mesh on an irregular domain is of great significance. In this paper, a novel unstructured mesh finite element method is developed for solving the time-space fractional wave equation on a two-dimensional irregular convex domain. The novel unstructured mesh Galerkin finite element method is used to discretize in space and the Crank-Nicholson scheme is used to discretize the Caputo time fractional derivative. The implementation of the unstructured mesh Crank-Nicholson Galerkin method (CNGM) is detailed and the stability and convergence of the numerical scheme are analyzed. Numerical examples are presented to verify the theoretical analysis. To highlight the ability of the proposed unstructured mesh Galerkin finite element method, a comparison of the unstructured mesh with the structured mesh in the implementation of the numerical scheme is conducted. The proposed numerical method using an unstructured mesh is shown to be more effective and feasible for practical applications involving irregular convex domains.

Soil salinization is a serious resource and ecological problem globally. The Weigan River–Kuqa River Delta Oasis is a key region in the arid and semi-arid regions of China with prominent soil salinization. The saline soils in the oasis are widely distributed over a large area, causing great harm to agricultural development and the environment. Remote sensing monitoring can provide a reference method for the management of regional salinization. We extracted the spectral indices and performed a correlation analysis using soil measurement data and Sentinel-2 remote sensing data. Then, two-dimensional feature space inversion models for soil salinity were constructed based on the preferred spectral indices, namely, the canopy response salinity index (CRSI), composite spectral response index (COSRI), normalized difference water index (NDWI), and green atmospherically resistant vegetation index (GARI). The soil salinity in a typical saline zone in the Weigan River–Kuqa River Delta Oasis was monitored and analyzed. We found that the inversion of the CRSI-COSRI model was optimal (R2 of 0.669), followed by the CRSI-NDWI (0.656) and CRSI-GARI (0.604) models. Therefore, a model based on the CRSI-COSRI feature space can effectively extract the soil salinization information for the study area. This is of great significance to understanding the salinization situation in the Weigan River–Kuqa River Delta Oasis, enriching salinization remote sensing monitoring methods, and solving the soil salinization problem in China.

The problem of finding discrete energy values of a particle with negative charge equal in absolute value to the elementary charge in one-dimensional space ( e → e 1 ), located in the field of a nucleus with charge ( Ze 1 ) with potential corresponding to the space of this dimension ( N = 1) and different from the potential of the nucleus in three-dimensional space ( N = 3) is solved in the quasiclassical approximation. For the one-dimensional case, the corresponding Schrödinger equation is solved and exact energy values are obtained, coincident with the quasiclassical approximation in the limit of large quantum numbers, and the wave function, expressed in terms of the Airy function, is found. In this latter approach, the energy values depend on the zeros of the Airy function. Considerations are discussed, touching on the possibility of solution of the Schrödinger equation for a hydrogen-like atom in two-dimensional space ( N = 2).