Towards a unified theory of fractional and nonlocal vector calculus

Nonlocal and fractional-order models capture effects that classical partial differential equations cannot describe; for this reason, they are suitable for a broad class of engineering and scientific applications that feature multiscale or anomalous behavior. This has driven a desire for a vector calculus that includes nonlocal and fractional gradient, divergence and Laplacian type operators, as well as tools such as Green’s identities, to model subsurface transport, turbulence, and conservation laws. In the literature, several independent definitions and theories of nonlocal and fractional vector calculus have been put forward. Some have been studied rigorously and in depth, while others have been introduced ad-hoc for specific applications. The goal of this work is to provide foundations for a unified vector calculus by (1) consolidating fractional vector calculus as a special case of nonlocal vector calculus, (2) relating unweighted and weighted Laplacian operators by introducing an equivalence kernel, and (3) proving a form of Green’s identity to unify the corresponding variational frameworks for the resulting nonlocal volume-constrained problems. The proposed framework goes beyond the analysis of nonlocal equations by supporting new model discovery, establishing theory and interpretation for a broad class of operators, and providing useful analogues of standard tools from the classical vector calculus.

Intelligent Control of Uncertain PMSM Based on Stable and Adaptive Discrete-Time Neural Network Compensators

In this paper, stable and adaptive neural network compensators are proposed to control the uncertain permanent magnet synchronous motor (PMSM). Firstly, the overall uncertainties caused by mathematical modelling, parameters variation during operation and external load torque disturbances are modelled. Secondly, a new motion control scheme, where (d-q) current loops are dotted by two on-line tuning neural network compensators (NNCs), is used to compensate these uncertainties. As a result, the speed control loop is processed easily by proportional integral (PI) controller. Stability of the closed-loop system is also designed according to the Lyapunov stability. Compared to classical vector control, the simulations of PMSM system at different speeds including nominal, low and high speed, with and without uncertainties, show the effectiveness of the proposed control scheme.

New Stable Non-Vector Control Structure for Induction Motor Drive

The article focuses on a design and experimental verification of continuous nonlinear systems control based on a new control structure based on a linear reference model. An application of Lyapunov’s second method ensures its asymptotic stability conditions. The basic idea in the development of the control structure consists of utilizing additional information from a newly introduced state variable. The structure is applied for angular speed control of an induction motor (IM) drive representing a higher-order nonlinear system. The developed control algorithm helps to achieve the zero steady-state control deviation of the IM drive angular speed. Simulations and experiments performed in various operating states of the IM drive confirm the advantages of the new control structure. Except for set dynamics, the method ensures that the system is stable, invariant to disturbances, and is robust against variations of the parameters. When comparing the obtained control structure of the IM control with the classical vector control, the proposed control structure is simpler. In addition, the proposed control structure is linear, robust against variation in important parameters and invariant against external disturbances. The main advantage over conventional control techniques consists of the fact that the controller design does not require any exact knowledge of the system parameters and, moreover, it does not suffer from system stability problems. The method will find a wide applicability not only in the field of AC controlled drives with IM but also generally in control of industry applications.

Soliton Domain Wall Concept: Analytical and Numerical Investigation in Digital Magnetic Recording System

To introduce Soliton theory in magnetic recording systems, we begin with the profile of Domain Wall, which is key elements of recording systems, knowing that many different Domain Walls shape exist. For this, we consider the recording media as chain of atoms (spin) and, we use the Hamiltonian to describe the global state of the system; by taking into consideration interaction between the neighboring spin and anisotropic interaction. Spins are considered as classical vector; for that, we defined the cosine and sine of angles that specify the position of the spins. They are developed in Taylor’s series until second order then using the approximation of continuous medium we obtained the Lagragian relation. This Lagragian enables us to describe the dynamics of spin through the wave velocity. As we are fine just the profile of domain wall it is beneficial for us to consider the wall at rest (static) and by the aid of Euler equation we obtain two simple equations; using the equilibrium conditions, the differential equation is obtained and solved by the quadratic method and separation variables method. The profile of domain wall that we obtain is at a particular position, then analytical and numerical simulation give us the opportunity to see that profile of that domain wall is a Kink, anti-Kink Soliton and also Soliton Train. Using this magnetic Soliton wave (Domain Wall), we also evaluate the playback voltage V (x), the peak voltage and the half pulse width PW50 to confirm the uses of this DW profile in magnetic recording systems and insure validity of this work.

Using Network Embedding to Obtain a Richer and More Stable Network Layout for a Large Scale Bibliometric Network

PurposeThe goal of this study is to explore whether deep learning based embedded models can provide a better visualization solution for large citation networks.Design/methodology/approachOur team compared the visualization approach borrowed from the deep learning community with the well-known bibliometric network visualization for large scale data. 47,294 highly cited papers were visualized by using three network embedding models plus the t-SNE dimensionality reduction technique. Besides, three base maps were created with the same dataset for evaluation purposes. All base maps used the classic OpenOrd method with different edge cutting strategies and parameters.FindingsThe network embedded maps with t-SNE preserve a very similar global structure to the full edges classic force-directed map, while the maps vary in local structure. Among them, the Node2Vec model has the best overall visualization performance, the local structure has been significantly improved and the maps’ layout has very high stability.Research limitationsThe computational and time costs of training are very high for network embedded models to obtain high dimensional latent vector. Only one dimensionality reduction technique was tested.Practical implicationsThis paper demonstrates that the network embedding models are able to accurately reconstruct the large bibliometric network in the vector space. In the future, apart from network visualization, many classical vector-based machine learning algorithms can be applied to network representations for solving bibliometric analysis tasks.Originality/valueThis paper provides the first systematic comparison of classical science mapping visualization with network embedding based visualization on a large scale dataset. We showed deep learning based network embedding model with t-SNE can provide a richer, more stable science map. We also designed a practical evaluation method to investigate and compare maps.

Quantum Bundles

A quantum bundle, namely a bundle whose sections are quantum fields, is defined as a classical vector bundle tensorialised by a suitable operator algebra related to a bundle of quantum states. The fundamental differential geometric notions for quantum bundles, including tangent, vertical and jet prolongations, and connections, can be conveniently introduced in terms F-smoothness. A short introduction to anti-fields and Batalin-Vilkovisky algebra naturally fits into this scheme.

Investigating Granger causality with state-space representation of time series: a case study of the total water storage anomaly over Australia

<p>Originating from econometrics, the concept of Granger causality (GC) has been widely used in a variety of fields, including climate sciences, to infer directional dependencies between stochastic variables.  Going one step further than the simple detection of lag-correlations, GC evaluates the directed interaction of a variable Y on a variable X by quantifying the improvement of prediction of future values of X when past values of Y are considered or omitted. Although not prescribed initially as such, GC is routinely computed from an estimated vector autoregressive model of the data of interest X, with and without the exogenous variable Y. However, such a modelling is somewhat restrictive and not suitable for filtered, sampled and noisy time series which may contain a moving-average component, impairing at the same time the quality of the GC estimator. Conversely, state-space representation offers a much more general framework for linear time series modelling.</p><p>In this study, we use Granger causality in the framework of a state-space modelling of time series to infer the presence of causal influences of the sea surface temperature (SST) and the 500hPa geopotential height on the Terrestrial Water Storage Anomaly (TWSA) over Australia[PS1] . A first and critical step is to reduce the high-dimension of the spatio-temporal data to a size compatible with classical state-space modelling algorithms. To do that we extract a limited number of leading modes of variability from the geophysical fields. Next, the state-space models of the extracted modes are identified using subspace-based methods. Then, the Granger causality of every mode of SST (resp. 500hPa geopotential height) on TWSA is estimated. Finally, we discuss the capability of the presented method to detect real directional dependencies in the light of current knowledge on Australia’s rainfall climatology and compare it to the results obtained with the classical vector autoregressive models.</p>

Theoretical and Experimental Study on Echo Fluctuation Suppression of a Cirrus Cloud by Millimeter Wave MIMO Radar

The scattering properties of nonspherical particles can be approximately computed by equivalent spherical theory. The scattering properties of ice particles were approximately computed by Rayleigh approximation because the sizes of the ice particles are smaller than the wavelength of millimeter wave radar. Based on the above assumption, the echo fluctuation of moving particles was analyzed by computing the total backscattering field of a cirrus cloud using the classical vector potential technique. The simulation results showed that echo fluctuation influences the accuracy of retrieving the physical parameters of a cloud. To suppress the echo fluctuation of moving ice particles, a video integrator of a millimeter wave cloud radar would be used. However, video integrators lose the rapidly changing information of ice particles and reduce radar range resolution; thus, we propose the pace-diversity technique of MIMO radar to reduce the echo fluctuation, which could be validated by theoretical computation and experimental measurements.