Generalized covariant differentiation and axiom-based tensor analysis

This paper reports the new progresses in the axiomatization of tensor anal- ysis, including the thought of axiomatization, the concept of generalized components, the axiom of covariant form invariability, the axiomatized definition, the algebraic structure, the transformation group, and the simple calculation of generalized covariant differentia- tions. These progresses strengthen the tendency of the axiomatization of tensor analysis.

A Novel Remote Sensing Signal De-noising Algorithm based on Neural Networks and Tensor Analysis

. This paper proposes a novel remote sensing signal de-noising algorithm based on neural networks and tensor analysis. The defects exist in a constant deviation between the wavelet coeffi cients and that the wavelet coefficients of the noisy signal to estimate the discontinuity of hard threshold function and soft threshold function, limiting its further application in order to overcome this shortcoming, this paper proposes a new threshold function, compared with the original threshold function, a new threshold function is simple and easy to calculate, not only with the soft threshold function is continuous. To deal with this drawback, we integrate the NN to enhance the model. Neural network belongs to the basic unsupervised learning of neural networks, the principle of competition based on the mechanism of learning and biological and the memory capacity can be increased as the number of learning patterns increases, not only offi ine learning can also be carried out on-line ＂learning while learning＂ type. The integrated algorithm can host better performance.

Block Incremental Dense Tucker Decomposition with Application to Spatial and Temporal Analysis of Air Quality Data

How can we efficiently store and mine dynamically generated dense tensors for modeling the behavior of multidimensional dynamic data?Much of the multidimensional dynamic data in the real world is generated in the form of time-growing tensors.For example,air quality tensor data consists of multiple sensory values gathered from wide locations for a long time.Such data,accumulated over time,is redundant and consumes a lot ofmemory in its raw form.We need a way to efficiently store dynamically generated tensor data that increase over time and to model their behavior on demand between arbitrary time blocks.To this end,we propose a Block IncrementalDense Tucker Decomposition(BID-Tucker)method for efficient storage and on-demand modeling ofmultidimensional spatiotemporal data.Assuming that tensors come in unit blocks where only the time domain changes,our proposed BID-Tucker first slices the blocks into matrices and decomposes them via singular value decomposition(SVD).The SVDs of the time×space sliced matrices are stored instead of the raw tensor blocks to save space.When modeling from data is required at particular time blocks,the SVDs of corresponding time blocks are retrieved and incremented to be used for Tucker decomposition.The factor matrices and core tensor of the decomposed results can then be used for further data analysis.We compared our proposed BID-Tucker with D-Tucker,which our method extends,and vanilla Tucker decomposition.We show that our BID-Tucker is faster than both D-Tucker and vanilla Tucker decomposition and uses less memory for storage with a comparable reconstruction error.We applied our proposed BID-Tucker to model the spatial and temporal trends of air quality data collected in South Korea from 2018 to 2022.We were able to model the spatial and temporal air quality trends.We were also able to verify unusual events,such as chronic ozone alerts and large fire events.

A survey of some tensor analysis techniques for biological systems

Background:Since biological systems are complex and often involve multiple types of genomic relationships,tensor analysis methods can be utilized to elucidate these hidden complex relationships.There is a pressing need for this,as the interpretation of the results of high-throughput experiments has advanced at a much slower pace than the accumulation of data.Results:In this review we provide an overview of some tensor analysis methods for biological systems.Conclusions:Tensors are natural and powerful generalizations of vectors and matrices to higher dimensions and play a fundamental role in physics,mathematics and many other areas.Tensor analysis methods can be used to provide the foundations of systematic approaches to distinguish significant higher order correlations among the elements of a complex systems via finding ensembles of a small number of reduced systems that provide a concise and representative summary of these correlations.

Acoustic emission characterization of microcracking in laboratory-scale hydraulic fracturing tests

Understanding microcracking near coalesced fracture generation is critically important for hydrocarbon and geothermal reservoir characterization as well as damage evaluation in civil engineering structures. Dense and sometimes random microcracking near coalesced fracture formation alters the mechanical properties of the nearby virgin material. Individual microcrack characterization is also significant in quantifying the material changes near the fracture faces （i.e. damage）. Acoustic emission （AE） monitoring and analysis provide unique information regarding the microcracking process temporally, and infor- mation concerning the source characterization of individual microcracks can be extracted. In this context, laboratory hydraulic fracture tests were carried out while monitoring the AEs from several piezoelectric transducers. In-depth post-processing of the AE event data was performed for the purpose of under- standing the individual source mechanisms. Several source characterization techniques including moment tensor inversion, event parametric analysis, and volumetric deformation analysis were adopted. Post-test fracture characterization through coring, slicing and micro-computed tomographic imaging was performed to determine the coalesced fracture location and structure. Distinct differences in fracture characteristics were found spatially in relation to the openhole injection interval. Individual microcrack AE analysis showed substantial energy reduction emanating spatially from the injection interval. It was quantitatively observed that the recorded AE signals provided sufficient information to generalize the damage radiating spatially away from the injection wellbore.

Extension of covariant derivative(Ⅰ): From component form to objective form

This paper extends the covariant derivative un der curved coordinate systems in 3D Euclid space. Based on the axiom of the covariant form invariability, the classical covariant derivative that can only act on components is ex tended to the generalized covariant derivative that can act on any geometric quantity including base vectors, vectors and tensors. Under the axiom, the algebra structure of the gen eralized covariant derivative is proved to be covariant dif ferential ring. Based on the powerful operation capabilities and simple analytical properties of the generalized covariant derivative, the tensor analysis in curved coordinate systems is simplified to a large extent.

Extension of covariant derivative(Ⅱ): From flat space to curved space

This paper extends the classical covariant deriva tive to the generalized covariant derivative on curved sur faces. The basement for the extension is similar to the pre vious paper, i.e., the axiom of the covariant form invariabil ity. Based on the generalized covariant derivative, a covari ant differential transformation group with orthogonal duality is set up. Through such orthogonal duality, tensor analy sis on curved surfaces is simplified intensively. Under the covariant differential transformation group, the differential invariabilities and integral invariabilities are constructed on curved surfaces.

Extension of covariant derivative(Ⅲ): From classical gradient to shape gradient

This paper further extends the generalized covari ant derivative from the first covariant derivative to the sec ond one on curved surfaces. Through the linear transforma tion between the first generalized covariant derivative and the second one, the second covariant differential transformation group is set up. Under this transformation group, the sec ond class of differential invariants and integral invariants on curved surfaces is made clear. Besides, the symmetric struc ture of the tensor analysis on curved surfaces are revealed.

Microstructural white matter lesion in Alzheimer's disease: a diffusion tensor imaging study using voxel-based analysis

Objective To study the microscopic changes of white matter and the relationship between white matter changes and cognitive impairment in Alzheimer’s disease(AD)using voxel-based analysis of DTI.Methods Thirty-seven patients with probable AD,and 32 normal controls(NC)were all examined by MMSE scores,and un-

Planning for selective amygdalohippocampectomy involving less neuronal fiber damage based on brain connectivity using tractography

Temporal lobe resection is an important treatment option for epilepsy that involves removal of potentially essential brain regions. Selective amygdalohippocampectomy is a widely performed temporal lobe surgery. We suggest starting the incision for selective amygdalohippocampectomy at the inferior temporal gyrus based on diffusion magnetic resonance imaging（MRI） tractography. Diffusion MRI data from 20 normal participants were obtained from Parkinson＇s Progression Markers Initiative（PPMI） database（www.ppmi-info.org）. A tractography algorithm was applied to extract neuronal fiber information for the temporal lobe, hippocampus, and amygdala. Fiber information was analyzed in terms of the number of fibers and betweenness centrality. Distances between starting incisions and surgical target regions were also considered to explore the length of the surgical path. Middle temporal and superior temporal gyrus regions have higher connectivity values than the inferior temporal gyrus and thus are not good candidates for starting the incision. The distances between inferior temporal gyrus and surgical target regions were shorter than those between middle temporal gyrus and target regions. Thus, the inferior temporal gyrus is a good candidate for starting the incision. Starting the incision from the inferior temporal gyrus would spare the important（in terms of betweenness centrality values） middle region and shorten the distance to the target regions of the hippocampus and amygdala.

Modeling Reynolds stress anisotropy invariants via machine learning

The presentation and modeling of turbulence anisotropy are crucial for studying large-scale turbulence structures and constructing turbulence models.However,accurately capturing anisotropic Reynolds stresses often relies on expensive direct numerical simulations(DNS).Recently,a hot topic in data-driven turbulence modeling is how to acquire accurate Reynolds stresses by the Reynolds-averaged Navier-Stokes(RANS)simulation and a limited amount of data from DNS.Many existing studies use mean flow characteristics as the input features of machine learning models to predict high-fidelity Reynolds stresses,but these approaches still lack robust generalization capabilities.In this paper,a deep neural network(DNN)is employed to build a model,mapping from tensor invariants of RANS mean flow features to the anisotropy invariants of high-fidelity Reynolds stresses.From the aspects of tensor analysis and input-output feature design,we try to enhance the generalization of the model while preserving invariance.A functional framework of Reynolds stress anisotropy invariants is derived theoretically.Complete irreducible invariants are then constructed from a tensor group,serving as alternative input features for DNN.Additionally,we propose a feature selection method based on the Fourier transform of periodic flows.The results demonstrate that the data-driven model achieves a high level of accuracy in predicting turbulence anisotropy of flows over periodic hills and converging-diverging channels.Moreover,the well-trained model exhibits strong generalization capabilities concerning various shapes and higher Reynolds numbers.This approach can also provide valuable insights for feature selection and data generation for data-driven turbulence models.

A TURBULENCE MODEL FOR VARYING DENSITY FLOW IN GENERAL CURVILINEAR COORDINATES

In this paper a three-dimensional turbulence model equation with irregular domain and variable density of incompressible flow in general curvilinear coordinates is developed by the tensor analysis. The equations can be conveniently and wildly used to solve problems in the field of hydraulics, environment and ocean engineering.