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Aiming at problems that in direction relation matrix model, recognition ability for distinguishing direction changes in the same cardinal direction is insufficient, the direction distance references for different cardinal directions are defined incompletely, and the distance calculation between arbitrarily direction relation matrices is not accurate enough, this paper proposes a direction relation two-tuple model, which combines grid-based direction relation matrix and centroid-based direction relation matrix to concern both distribution ratio variations and centroid position variations for objects, thus distinguishing the direction difference in the same cardinal direction. Meanwhile, the traditional neighborhood graph is optimized based on human spatial cognition, and a centroid direction distance reference suitable for arbitrarily direction relationships is established. Finally the Earth mover's distance (EMD) is utilized to further improve the accuracy of distance calculation between direction relation two-tuples. Experiments indicate the method is simple and feasible, the measurement results are more consistent with human cognition, and can be better applied to tasks like cartographic generalization results evaluation.

Describing and understanding the basement relief of sedimentary basins is vital for oil and gas exploration. The traditional method to map an interface in each spatial direction is based on 3D modeling of gravity Bouguer anomalies with variable lateral and vertical density contrasts using a priori information derived from other types of geoscience data sets as constraints (e.g., well and/or seismic data). However, in the preexploration stage, vertical gravity [Formula: see text], which is sometimes the only available geophysical data, is typically used to recover smooth density contrast distributions under a generic set of constraints. Apparently, the use of the [Formula: see text] component is not sufficient to produce geologically reasonable interpretations with high resolution. To address this, we have developed a novel process of hybrid inversion, combining gravity migration and inversion using the same [Formula: see text] data set, to distinguish the complicated interface between basement and sedimentary basin rocks from a full-space inverted density distribution volume. First, a 3D-migrated model delineating the basic sedimentary basin structure is derived using a focusing gravity iterative migration method, where a priori information is not necessary. Subsequently, under the framework of the regularized focusing conjugate inversion algorithm, a high-resolution density contrast model is inverted for delineation of the basement boundary by integrating the 3D-migrated density model as a priori information. We examine the method using one synthetic example and a field data case, of which a transformed resolution density matrix is developed from logarithmic space to qualitatively evaluate the practical resolutions. The high resolution of the density distribution of the Cretaceous basement with a clear interface is achieved and verified by limited seismic data and strata markers in limited wells.

In this article we propose fully discrete least-squares spectral element method for parabolic interface problems in R 2 . Crank–Nicolson scheme is used in time and higher order spectral elements are used in the spatial direction. This method is based on the nonconforming spectral element method proposed in Kishore Kumar and Naga Raju (2012). The proposed method is least-squares spectral element method. Nonconforming higher order spectral elements have been used. The jump in the solution and its normal derivative across the interface are enforced (in an appropriate Sobolev norm) in the minimizing functional. The method is second order accurate in time and exponentially accurate in spatial direction with p − version in L 2 ( H 1 ) norm. Numerical results are presented to show the efficiency of the proposed method.

The nonlinear time-fractional stochastic fourth-order reaction–diffusion equation perturbed by noises is considered by the mixed finite element method in this paper. Based on the mixed finite element in spatial direction and the generalized BDF 2 − θ in temporal discretization, the semi- and fully-discrete schemes are obtained. Further, the error estimates for the semi- and fully-discretizations and the regularity of solution are derived by some important lemmas. The numerical experiment is conducted to further confirm the theoretical results.

Analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media has been investigated extensively in the last several decades. Of particular interest in practical applications is the lowest-order Galerkin-mixed method, in which a linear Lagrange FE approximation is used for the concentration and the lowest-order Raviart-Thomas FE approximation is used for the velocity/pressure. The previous works only showed the first-order accuracy of the method in L 2 L^2 -norm in spatial direction, which however is not optimal and valid only under certain extra restrictions on both time step and spatial mesh. In this paper, we provide new and optimal L 2 L^2 -norm error estimates of Galerkin-mixed FEMs for all three components in a general case. In particular, for the lowest-order Galerkin-mixed FEM, we show unconditionally the second-order accuracy in L 2 L^2 -norm for the concentration. Numerical results for both two- and three-dimensional models are presented to confirm our theoretical analysis. More important is that our approach can be extended to the analysis of mixed FEMs for many strongly coupled systems to obtain optimal error estimates for all components.

Increasing land demand has resulted land conversion, one of which is forest land conversion. The objectives of this research are to identify: the conversion patterns of forest land use, the classification of land capability, the spatial direction of agroforestry programsand the land conversion in 2025. This research implements map overlay, land capability analysis, and CA-Markov method. The results show that the pattern the conversion in 1990-2015is dynamic, the biggest change was from 1996 to 2003 covering the area of 24,487 hectares. There are five classesof Land capability: classes II, III, IV, VI, and VII with erosion as the biggest limiting factor (e4) dominant inclasses IV and VI. Management suggestions consist of 11,361 hectares of agroforestry patterns, 11,228 hectaresof reforestation, 3,013 hectares of Tanah Obyek Reformasi Agraria, 102,928 hectares of other management inaccordance with Rencana Karya Usaha Pemanfaatan Hasil Hutan Kayu and Rencana Pengelolaan Hutan JangkaPanjang. Prediction of land use change in 2025 with bussiness as ussual scenario shows 3.6% reduction of HutanSekunder with RPHJP implementation scenario, 0.5% HS is reduced; with the strategy of agroforestry scenarioshows 105.3% HS increase.

In this paper, a fully discrete Local discontinuous Galerkin (LDG) method is discussed for a time-fractional reaction diffusion initial-boundary value problem with discontinuous diffusive coefficient. In this method the well-known L2- 1 σ scheme on graded meshes is presented to deal with the weak singularity at initial time t = 0 , while in the spatial direction a LDG method on a uniform mesh is presented to handle the discontinuous coefficient. By the discrete fractional Gronwall inequality, the L 2 -norm stability and consistency estimate results are derived for the proposed fully discrete LDG method. Numerical experiments are presented to verify the sharpness of the error analysis.

In this paper, two classes of finite element methods (FEMs) for the two-dimensional time fractional diffusion equation (TFDE) with non-smooth solution are proposed and analyzed. In the temporal direction, we adopt nonuniform L2- 1 σ method, and in the spatial direction, the conforming bilinear element and the nonconforming modified quasi-Wilson element are utilized. For the proposed conforming and nonconforming FEMs, by using new fractional discrete Gronwall inequalities, the theoretical analysis including the L 2 -norm error estimates and H 1 -norm superclose results are given in details. Furthermore, by virtue of the interpolated postprocessing techniques, the global H 1 -norm superconvergence results are presented. Finally, some numerical results illustrate the correctness of theoretical analysis.

Seismic trace interpolation is an important technique because irregular or insufficient sampling data along the spatial direction may lead to inevitable errors in multiple suppression, imaging, and inversion. Many interpolation methods have been studied for irregularly sampled data. Inspired by the working idea of the autoencoder and convolutional neural network, we have performed seismic trace interpolation by using the convolutional autoencoder (CAE). The irregularly sampled data are taken as corrupted data. By using a training data set including pairs of the corrupted and complete data, CAE can automatically learn to extract features from the corrupted data and reconstruct the complete data from the extracted features. It can avoid some assumptions in the traditional trace interpolation method such as the linearity of events, low-rankness, or sparsity. In addition, once the CAE network training is completed, the corrupted seismic data can be interpolated immediately with very low computational cost. A CAE network composed of three convolutional layers and three deconvolutional layers is designed to explore the capabilities of CAE-based seismic trace interpolation for an irregularly sampled data set. To solve the problem of rare complete shot gathers in field data applications, the trained network on synthetic data is used as an initialization of the network training on field data, called the transfer learning strategy. Experiments on synthetic and field data sets indicate the validity and flexibility of the trained CAE. Compared with the curvelet-transform-based method, CAE can lead to comparable or better interpolation performances efficiently. The transfer learning strategy enhances the training efficiency on field data and improves the interpolation performance of CAE with limited training data.

The structure of neuronal dendritic trees plays a key role in the integration of synaptic inputs in neurons. Therefore, characterization of the morphology of dendrites is essential for a better understanding of neuronal function. However, the complexity of dendritic trees, both when isolated and especially when located within neuronal networks, has not been completely understood. We developed a new computational tool, SOA (Segmentation and Orientation Analysis), which allows automatic measurement of the orientation of dendritic branches from fluorescence images of 2D neuronal cultures. SOA, written in Python, uses segmentation to distinguish dendritic branches from the image background and accumulates a database on the spatial direction of each branch. The database is then used to calculate morphological parameters such as the directional distribution of dendritic branches in a network and the prevalence of parallel dendritic branch growth. The data obtained can be used to detect structural changes in dendrites in response to neuronal activity and to biological and pharmacological stimuli.

Located deep in the temporal bone, the semicircular canal is a subtle structure that requires a spatial coordinate system for measurement and observation. In this study, 55 semicircular canal and eyeball models were obtained by segmentation of MRI data. The spatial coordinate system was established by taking the top of the common crus and the bottom of the eyeball as the horizontal plane. First, the plane equation was established according to the centerline of the semicircular canals. Then, according to the parameters of the plane equation, the plane normal vectors were obtained. Finally, the average unit normal vector of each semicircular canal plane was obtained by calculating the average value of the vectors. The standard normal vectors of the and left posterior semicircular canal, superior semicircular canal and lateral semicircular canal were [−0.651, 0.702, 0.287], [0.749, 0.577, 0.324], [−0.017, −0.299, 0.954], [0.660, 0.702, 0.266], [−0.739, 0.588, 0.329], [0.025, −0.279, 0.960]. The different angles for the different ways of calculating the standard normal vectors of the right and left posterior semicircular canal, superior semicircular canal and lateral semicircular canal were 0.011, 0.028, 0.008, 0.011, 0.024, and 0.006 degrees. The technology for measuring the semicircular canal spatial attitudes in this study are reliable, and the measurement results can guide vestibular function examinations and help with guiding the diagnosis and treatment of BPPV.

In this paper, we consider the nonconforming virtual element method (VEM) for the approximation of the time fractional reaction–subdiffusion equation involving the Caputo fractional derivative. For the numerical discrete method of the Caputo fractional derivative, we permit the use of nonuniform time steps, since they are helpful to deal with the non-smooth system. Meanwhile, the nonconforming VEM, which is constructed for any order of accuracy and for very general shaped polygonal and polyhedral meshes, is adopted for the discretization of the spatial direction. By introducing a new Ritz projection operator and using two extended ty$$L^2$$-normpes of continuous and discrete fractional Gronwall inequalities, the optimal error estimates for the spatial semi-discrete and temporal-spatial fully discrete systems are proved detailedly. Besides, the fully discrete scheme is proved to be unconditionally stable with regard to the $$L^2$$- and $$H^1$$-norms, respectively. Finally, some numerical calculations are implemented to verify the theoretical results.

We study the spreading speeds of a species described by a reaction–diffusion equation with a shifting habitat on which the species’ growth rate increases in the positive spatial direction. Persistence and the rightward spreading speed have been previously investigated in the case that the species grows near positive infinity and declines near negative infinity. In the present paper we determine the leftward spreading speed for this case and both the rightward and leftward spreading speeds for the case that the species grows everywhere in space. We show that the spreading speeds depend on c, the speed at which the habitat shifts, and numbers $$c^*(\infty )$$ and $$c^*(-\infty )$$ given by the diffusion coefficient and species growth rates at $$\infty $$ and $$-\infty $$ respectively. We demonstrate that if the species declines near negative infinity, the leftward spreading speed is $$\min \{-c, c^*(\infty )\}$$ , and if species grows everywhere, under appropriate conditions, the rightward spreading speed is either $$c^*(\infty )$$ or $$c^*(-\infty )$$ , and the leftward spreading speed is one of $$c^*(\infty )$$ , $$c^*(-\infty )$$ and $$-c$$ . Our results show that it is possible for a solution to form a two-layer wave, with the propagation speeds analytically determined.

In this paper, we consider the numerical solution of the two dimensional fractional Stokes’ first problem for a heated generalized second grade fluid. The proposed method is based on the L1 finite difference scheme for the temporal direction while the Legendre spectral method for the spatial direction. Numerical stability and convergence of the method are rigorously established. At the mean time, the case of non-smooth solution is considered by adding some correction terms. Numerical experiments are included, which verify the theoretical predictions.

In this note we perform canonical analysis of T-duality for non-relativistic string in stringy Newton-Cartan background. We confirm recent result that T-duality along longitudinal spatial direction of stringy Newton-Cartan geometry maps non-relativistic string to the relativistic string that propagates on the background with light-like isometry., Comment: 12 pages, references added

Early postoperative rehabilitation training for supracondylar fracture of femur aids in accelerating healing with shorter recovery periods. Presently, clinical studies on early postoperative weight training are still in the nascent stage. The weight-bearing capacity at different healing stages typically depends on clinical experience, and there is a lack of standards to quantify the weight that is conducive to healing of fractures. In this paper, a three-dimensional (3D) geometric model of the femur is obtained using imaging data, a locking plate fixation model of a simple supracondylar fracture of the femur, considering the angle and spatial direction of the fracture surface, is established, the stress distribution and load transmission mechanism of the fracture fixation model in a single-leg standing posture are studied, and the weight-bearing capacity of a standing single leg at the early stage of fracture is given. This provides the basis for objective quantification of early postoperative weight-bearing capacity.

It is well known that reaction-diffusion systems can describe the interactions of different species in ecological systems. In this work, we propose a numerical method for solving a general two-species reaction-diffusion system. The method comprises a standard finite difference to discretize in time direction and Sinc collocation method in spatial direction. A series of numerical experiments demonstrate the accuracy and good performance of the algorithm. The biological significance of the numerical results was also discussed and presented in figures.

This paper introduces new fully implicit numerical schemes for solving 1D and 2D unsteady Burgers' equation. The non-linear Burgers' equation is discretized in the spatial direction by using second order Finite difference method which converts the Burgers' equation to non-linear system of ODEs. Then, the backward differentiation formula of order two (BDF-2) is employed to march the solution in the time direction. The non-linear term in the obtained system is linearized without any transformation; and it forms a system of linear algebraic equations that is solved by using Thomas's algorithm. Accuracy and performance of the proposed schemes are studied by using four test problems with Dirichlet and Neumann boundary conditions. Comparing the numerical results with exact solutions and the solutions of other schemes shows that the proposed schemes are simple, efficient and accurate even for the cases with high Reynolds numbers.

Numerical mode matching (NMM) methods are widely used for analyzing wave propagation and scattering in structures that are piece-wise uniform along one spatial direction. For open structures that are unbounded in transverse directions (perpendicular to the uniform direction), the NMM methods use the perfectly matched layer (PML) technique to truncate the transverse variables. When incident waves are specified in homogeneous media surrounding the main structure, the total field is not always outgoing, and the NMM methods rely on reference solutions for each uniform segment. Existing NMM methods have difficulty handing gracing incident waves and special incident waves related to the onset of total internal reflection, and are not very efficient at computing reference solutions for non-plane incident waves. In this paper, a new NMM method is developed to overcome these limitations. A Robin-type boundary condition is proposed to ensure that non-propagating and non-decaying wave field components are not reflected by truncated PMLs. Exponential convergence of the PML solutions based on the hybrid Dirichlet-Robin boundary condition is established theoretically. A fast method is developed for computing reference solutions for cylindrical incident waves. The new NMM is implemented for two-dimensional structures and polarized electromagnetic waves. Numerical experiments are carried out to validate the new NMM method and to demonstrate its performance., 26 pages, 15 figures

We investigate the phase behaviour of a two-dimensional colloidal model system of ultra-soft particles on a substrate which varies periodically along one spatial direction. Our calculations are based on mean-field density functional theory for a system of particles interacting via an ultra-soft potential, that is, the generalized exponential model with index four (\mbox{GEM-4}). For suitable substrate periodicities (with commensurability parameter $p=2$), we find a succession of phase transitions from a modulated liquid to a locked smectic and then to a locked floating solid phase. The appearance of a locked smectic phase is consistent with earlier theoretical predictions and experiments for freezing of more repulsive systems on structured surfaces (with $p=2$). However, the present ultra-soft system does not display re-entrant melting. We here investigate the details of the density distributions of the different phases, thereby supplementing earlier work on GEM-4 systems with $p=1$ [Phys. Rev. E \textbf{101}, 012609 (2020)]. Interestingly, the observed succession of phase transitions can be triggered through different paths along which physical control parameters are changed., This is an original manuscript of an article published by Taylor & Francis in Molecular Physics on 22 Jan 2021, available online: http://www.tandfonline.com/10.1080/00268976.2021.1875078

Robot systems will soon be able to hand over objects to humans as well as receive objects from humans in a robust way. From an ergonomics point of view, it is required to evaluate those robot systems and their interactions with humans based on appropriate parameters and design them accordingly. Therefore, we conducted an experiment with human-to-human hand-overs. The aim was to analyze different conditions of hand-overs that occur in our daily life such as different spatial direction, cups with varying filling quantities and varying states of perception of the receiving person. It was shown that cups with a higher filling level lead to a significantly higher duration of the interaction phase than cups with a lower filling level. Additionally, perceptual impairment of the receiver and thereby a lack of feedback led to a higher duration of the interaction phase.

This paper describes a practical application of a recently developed and validated high-fidelity antenna model. The paper first derives a method of populating the high-fidelity antenna model’s parameters to achieve a specific total gain and polarization in a single spatial direction. The paper then demonstrates the method’s validity with a successful application to a realistic example problem.

We apply the charge pumping argument to fermionic tensor network representations of d-dimensional topological insulators (TIs) to obtain tensor network states for (d+1)-dimensional TIs. We exemplify the method by constructing a two-dimensional projected entangled pair states (PEPS)for a Chern insulator starting from a matrix product state (MPS) in d=1 describing pumping in the Su-Schrieffer-Heeger (SSH) model. In extending the argument to second-order TIs, we build a three-dimensional TNS for a chiral hinge TI from a PEPS in d=2 for the obstructed atomic insulator (OAI) of the quadrupole model. The (d+1)-dimensional TNSs obtained in this way have a constant bond dimension inherited from the d-dimensional TNSs in all but one spatial direction, making them candidates for numerical applications. From the d-dimensional models, we identify gapped next-nearest neighbour Hamiltonians interpolating between the trivial and OAI phases of the fully dimerized SSH and quadrupole models, whose ground states are given by an MPS and a PEPS with a constant bond dimension equal to 2, respectively., 36 pages, 10 figures

Based on the finite-element method (FEM) in the spatial direction and L1-CN approximation in the temporal direction, respectively, numerical analyses are proposed for 2D two-term mixed time-fractional diffusion-wave equations. First, a fully discrete approximate scheme is established for the equation, and it is proved to be unconditionally stable. Then, rigorous proofs are provided for the convergence result in the $L^2$- norm and superclose properties in $H^1$- norm with order $~O(h^2+\displaystyle\tau^{\min\{2-\alpha_1,~3-\alpha\}})~~(0

Precise locations for the complete feather molt of North American landbirds, relative to their breeding territories, remain undocumented for most species. We analyzed >760,000 records of 140 species at 936 bird-capture stations to assess probabilities of recording both molting landbirds at their breeding sites and breeding birds at molting sites, and to investigate latitudinal, longitudinal, and elevational shifts from breeding to molting grounds. We demonstrate widespread evidence for molt-migrations among a variety of North American landbirds, including many migratory species previously thought to molt on “breeding grounds.” Geospatial differences between breeding and molting grounds were detected in all 4 compass directions as well as both upslope and downslope in elevation, while individuals of some species appeared to disperse to specific molting locations not discriminated by spatial direction or elevation from breeding territories. Although western North American species and populations ar...

Boundary integral method has been implemented successfully in practice for simulating problems with free boundaries. Though the method produces accurate and efficient numerical results, its convergence study is usually limited to numerical demonstrations by successively reducing time step and increasing resolution for a test problem. In this paper, we present a rigorous convergence and error analysis of the boundary integral method for a free boundary system. We focus our study on a nonlinear tumor growth problem. The boundary integral formulation yields a Fredholm type integral equation with moving boundaries. We show that in two dimensions, the convergence of the scheme in the L ∞ norm has first order accuracy on the time direction and Δ θ α on the spatial direction.

The fourth-order diffusion systems depict the wave and photon propagation in intense laser beams and play an important role in the phase separation in binary mixture. In this paper, by using orthogonal spline collocation (OSC) method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established for a class of fourth-order fractional reaction–diffusion equations. For the original unknown u and auxiliary variable v = Δ u , the full-discrete unconditional stabilities based on a priori analysis are derived by virtue of properties of OSC. Moreover, the convergence rates in L 2 -norm for unknown u are strictly investigated. At the same time, the optimal error estimates in H 1 -norm for unknown u and in L 2 -norm for variable v , are also derived, respectively. For further verifying the theoretical analysis, some numerical examples are provided.

In this paper, the superclose and superconvergence analysis of the nonlinear time-fractional thermistor problem are investigated by bilinear finite element method (FEM) for a fully-discrete scheme, in which the Caputo derivative is approximated by the classical L1 method. By dealing with the error estimates in the spatial direction rigorously, which are one order higher than the traditional FEMs, the superclose estimates in H1-norm are obtained for the corresponding variables based on the special properties of this element together with mean value technique. Subsequently, the global superconvergence results are derived by employing the interpolation postprocessing approach. Finally, a numerical experiment is carried out to confirm the theoretical analysis.

In the current manuscript, we consider a fractional partial integro-differential equation that has some applications in the electroanalytical chemistry. The fractional derivative is based on the Riemann–Liouville fractional integral. The current numerical investigation is based on the following procedures: at first, a difference scheme has been used to discrete the temporal direction, second, the local RBF-DQ method is employed to discrete the spatial direction, and finally, these procedures are combined to obtain a full-discrete scheme. For the constructed numerical technique, we prove the unconditional stability and also obtain an error bound. We employ some test problems to show the accuracy of the proposed technique. In addition, we compare the obtained numerical results using the present method with the existing methods in the literature.

The biomechanical changes in the spinal column are considered to be the main responsible for rachialgia. Although radiological techniques use ionizing radiation, they are the most applied tools to assess the biomechanics of the spine. To face this problem, non-invasive techniques must be developed. Vertebral Metrics is an ionizing radiation-free instrument designed to detect the 3D position of each vertebrae in a standing position. Using a stereo vision system combined with low intensity UV light, recognition is achieved with software capable of distinguishing fluorescent marks. The fluorescent marks are the skin projection of the vertex of the spinal processes. This paper presents a major development of Vertebral Metrics and its evaluation. It performs a scan in less than 45 s with a resolution on the order of 1 mm, in each spatial direction, therefore, allowing an accurate analysis of the spine. The instrument was applied to patients without associated pathology. Statistically significant differences between consecutive scans were not found. A positive correlation between the 3D positions of each vertebra and the homologous position of the other vertebrae was observed. Using Vertebral Metrics, innovative results can be obtained. It can be used in areas such as orthopedics, neurosurgery, and rehabilitation. Graphical abstract ᅟ.

This paper is concerned with mass and energy recovery by some conserved compact finite difference schemes for the nonlinear Schrödinger-Poisson equations. The mass and energy conservation, the unique solvability, convergence and stability of the proposed schemes are proved. It is shown that the proposed methods are of order 2 in temporal direction and order 4 in spatial direction. Numerical experiments are presented to illustrate our theoretical results.

In this paper, we derived a new compact finite difference scheme in the spatial direction and used the semi-implicit Euler-Maruyama approach in the temporal direction to study a stochastic extended Fisher-Kolmogorov equation with multiplicative noise numerically. Moreover, the analysis of consistency for the stochastic difference scheme was discussed and the stability analysis was proven in the mean square sense and by Fourier analysis. This approach is numerically analyzed to show the effect of random fluctuations occurring in nature and missing from the deterministic version of the equation and this illustrated in a numerical experiment.

A decline in the cognitive ability of mental rotation causes a poor sense of spatial direction and environmental cognitive capacity. Currently, training tasks for the elderly thus affected are still presented in 2D form. However, clinical research indicates that this strategy generates a cognitive load that reduces the interest of the trainees and diminishes the effects of training. In contrast, augmented reality (AR) is a rising solution that effectively reduces cognitive load, improves the sense of spatial direction of the elderly, and helps increase interest in training. We recruited 28 elderly (age65 years) for this study. Fourteen were randomly assigned to an active Intervention group and were given AR-based 3D hologram (AR-3DH) mental rotation training, and 14 were assigned to a group that used the traditional 2D model. Both groups took ABA-designed pre-and post-tests that required inferring the rotating shapes' states from standard mental rotation tasks. Change scores for the two groups were compared using error rates and reaction times as covariates. After six-week of training, the mental rotation ability of the Intervention group improved through the use of AR-3DH training system during the intervention phase. The practical and developmental implications of the findings are discussed. AR-3DH system can effectively reduces cognitive load and helps increase interest in training.AR-3DH system can improves participants' sense of spatial direction.AR-3DH system improved elderly participants' mental rotation ability after 3D game training.

In recent years, topological photonics has emerged as an exciting research field in which topological concepts, originally introduced to understand the behaviour of electrons in solid-state materials, are instead exploited to design and control the behaviour of light [1]. In this direction, the development of so-called "synthetic dimensions" offers a particularly powerful approach to exploring topological photonics. The central idea of this method is to identify and couple together a set of internal degrees of freedom so as to simulate the motion of a quantum particle along an extra spatial direction [2]. In this framework, a system with D-real dimensions can be made to behave like a D + d-dimensional system, when d synthetic dimensions are added to the system. This approach, therefore, both raises the prospect of simulating exotic systems with four or more spatial dimensions, but also provides, through the external control of couplings, a straightforward way to design topological lattices in different dimensions [3].