三维稳态Q-tensor液晶流系统各向异性的Liouville型定理

考虑速度分量的各向异性进行能量估计,得到三维稳态Q-tensor液晶流系统的Liouville型定理,即若u∈L^(q)(R^(3))∩˙H^(1)(R^(3)),u_(i)∈L xi q/q−2 L s xei(R×R^(2))(i=1,2,3),且Q∈H^(2)(R^(3)),其中2/q+1/s≥1/2,1≤s≤∞,2<q<∞,则该稳态系统只有平凡解.这个结论推广了已有的结果.

Euler Product Expressions of Absolute Tensor Products of Dirichlet L-Functions

In this paper, we calculate the absolute tensor square of the Dirichlet L-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a series which has the factors of the absolute tensor product of the Dirichlet L-functions. This study is a generalization of Akatsuka’s theorem on the Riemann zeta function, and gives a proof of Kurokawa’s prediction proposed in 1992.

Fault Diagnosis Scheme for Railway Switch Machine Using Multi-Sensor Fusion Tensor Machine

Railway switch machine is essential for maintaining the safety and punctuality of train operations.A data-driven fault diagnosis scheme for railway switch machine using tensor machine and multi-representation monitoring data is developed herein.Unlike existing methods,this approach takes into account the spatial information of the time series monitoring data,aligning with the domain expertise of on-site manual monitoring.Besides,a multi-sensor fusion tensor machine is designed to improve single signal data’s limitations in insufficient information.First,one-dimensional signal data is preprocessed and transformed into two-dimensional images.Afterward,the fusion feature tensor is created by utilizing the images of the three-phase current and employing the CANDE-COMP/PARAFAC(CP)decomposition method.Then,the tensor learning-based model is built using the extracted fusion feature tensor.The developed fault diagnosis scheme is valid with the field three-phase current dataset.The experiment indicates an enhanced performance of the developed fault diagnosis scheme over the current approach,particularly in terms of recall,precision,and F1-score.

Stress tensor determination by modified hydraulic tests on pre-existing fractures:Method and stress constraints

The hydraulic testing of pre-existing fractures(HTPF)is one of the most promising in situ stress measurement methods,particularly for three-dimensional stress tensor determination.However,the stress tensor determination based on the HTPF method requires at least six tests or a minimum of 14-15 tests(under different conditions)for reliable results.In this study,we modified the HTPF method by considering the shear stress on each pre-existing fracture,which increased the number of equations for the stress tensor determination and decreased the number of tests required.Different shear stresses were attributed to different fractures by random sampling;therefore,the stress tensors were obtained by searching for the optimal solution using the least squares criterion based on the Monte Carlo method.Thereafter,we constrained the stress tensor based on the tensile strength criterion,compressive strength criterion,and vertical stress constraints.The inverted stress tensors were presented and analyzed based on the tensorial nature of the stress using the Euclidean mean stress tensor.Two stress-measurement campaigns in Weifang(Shandong Province,China)and Mercantour road tunnel(France)were implemented to highlight the validity and efficiency of the modified HTPF(M-HTPF)method.The results showed that the M-HTPF method can be applied for stress tensor inversion using only three to four tests on pre-existing fractures,neglecting the stress gradient.The inversion results were confined to relatively small distribution dispersions and were significantly reliable and stable due to the shear stresses on the fractures and the stress constraints employed.The M-HTPF method is highly feasible and efficient for complete stress tensor determination in a single borehole.

Quantum geometric tensor and the topological characterization of the extended Su-Schrieffer-Heeger model

We investigate the quantum metric and topological Euler number in a cyclically modulated Su-Schrieffer-Heeger(SSH)model with long-range hopping terms.By computing the quantum geometry tensor,we derive exact expressions for the quantum metric and Berry curvature of the energy band electrons,and we obtain the phase diagram of the model marked by the first Chern number.Furthermore,we also obtain the topological Euler number of the energy band based on the Gauss-Bonnet theorem on the topological characterization of the closed Bloch states manifold in the first Brillouin zone.However,some regions where the Berry curvature is identically zero in the first Brillouin zone result in the degeneracy of the quantum metric,which leads to ill-defined non-integer topological Euler numbers.Nevertheless,the non-integer"Euler number"provides valuable insights and an upper bound for the absolute values of the Chern numbers.

Implementing and evaluating an automatic centroid moment tensor procedure for the Indonesia region and surrounding areas

The purpose of this research was to suggest an applicable procedure for computing the centroid moment tensor(CMT)automatically and in real time from earthquakes that occur in Indonesia and the surrounding areas.Gisola software was used to estimate the CMT solution by selecting the velocity model that best suited the local and regional geological conditions in Indonesia and the surrounding areas.The data used in this study were earthquakes with magnitudes of 5.4 to 8.0.High-quality,real-time broadband seismographic data were provided by the International Federation of Digital Seismograph Networks Web Services(FDSNWS)and the European Integrated Data Archive(EIDA)Federation in Indonesia and the surrounding areas.Furthermore,the inversion process and filter adjustment were carried out on the seismographic data to obtain good CMT solutions.The CMT solutions from Gisola provided good-quality solutions,in which all earthquake data had A-level quality(high quality,with good variant reduction).The Gisola CMT solution was justified with the Global CMT(GCMT)solution by using the Kagan angle value,with an average of approximately 11.2°.This result suggested that the CMT solution generated from Gisola was trustworthy and reliable.The Gisola CMT solution was typically available within approximately 15 minutes after an earthquake occurred.Once it met the quality requirement,it was automatically published on the internet.The catalog of local and regional earthquake records obtained through this technology holds great promise for improving the current understanding of regional seismic activity and ongoing tectonic processes.The accurate and real-time CMT solution generated by implementing the Gisola algorithm consisted of moment tensors and moment magnitudes,which provided invaluable insights into earthquakes occurring in Indonesia and the surrounding areas.

A Novel Tensor Decomposition-Based Efficient Detector for Low-Altitude Aerial Objects With Knowledge Distillation Scheme

Unmanned aerial vehicles(UAVs) have gained significant attention in practical applications, especially the low-altitude aerial(LAA) object detection imposes stringent requirements on recognition accuracy and computational resources. In this paper, the LAA images-oriented tensor decomposition and knowledge distillation-based network(TDKD-Net) is proposed,where the TT-format TD(tensor decomposition) and equalweighted response-based KD(knowledge distillation) methods are designed to minimize redundant parameters while ensuring comparable performance. Moreover, some robust network structures are developed, including the small object detection head and the dual-domain attention mechanism, which enable the model to leverage the learned knowledge from small-scale targets and selectively focus on salient features. Considering the imbalance of bounding box regression samples and the inaccuracy of regression geometric factors, the focal and efficient IoU(intersection of union) loss with optimal transport assignment(F-EIoU-OTA)mechanism is proposed to improve the detection accuracy. The proposed TDKD-Net is comprehensively evaluated through extensive experiments, and the results have demonstrated the effectiveness and superiority of the developed methods in comparison to other advanced detection algorithms, which also present high generalization and strong robustness. As a resource-efficient precise network, the complex detection of small and occluded LAA objects is also well addressed by TDKD-Net, which provides useful insights on handling imbalanced issues and realizing domain adaptation.

Least Squares One-Class Support Tensor Machine

One-class classification problem has become a popular problem in many fields, with a wide range of applications in anomaly detection, fault diagnosis, and face recognition. We investigate the one-class classification problem for second-order tensor data. Traditional vector-based one-class classification methods such as one-class support vector machine (OCSVM) and least squares one-class support vector machine (LSOCSVM) have limitations when tensor is used as input data, so we propose a new tensor one-class classification method, LSOCSTM, which directly uses tensor as input data. On one hand, using tensor as input data not only enables to classify tensor data, but also for vector data, classifying it after high dimensionalizing it into tensor still improves the classification accuracy and overcomes the over-fitting problem. On the other hand, different from one-class support tensor machine (OCSTM), we use squared loss instead of the original loss function so that we solve a series of linear equations instead of quadratic programming problems. Therefore, we use the distance to the hyperplane as a metric for classification, and the proposed method is more accurate and faster compared to existing methods. The experimental results show the high efficiency of the proposed method compared with several state-of-the-art methods.

关于Focal Loss的Tensor Train多项式分类器

在模式分类领域,多项式分类器因其复杂决策边界能力而得到广泛研究。利用TensorTrain分解形式来表示多项式分类器,可有效克服维数灾难。针对多项式分类器在训练过程中遇到的训练集分布不平衡问题,文章使用FocalLoss重塑了标准交叉损失,以降低分配给易分类样本的损失的权重,并在被广泛使用的图像分类数据集MNIST上验证了分类器的有效性。

Verification of neutron-induced fission product yields evaluated by a tensor decompsition model in transport-burnup simulations

Neutron-induced fission is an important research object in basic science.Moreover,its product yield data are an indispensable nuclear data basis in nuclear engineering and technology.The fission yield tensor decomposition(FYTD)model has been developed and used to evaluate the independent fission product yield.In general,fission yield data are verified by the direct comparison of experimental and evaluated data.However,such direct comparison cannot reflect the impact of the evaluated data on application scenarios,such as reactor transport-burnup simulation.Therefore,this study applies the evaluated fission yield data in transport-burnup simulation to verify their accuracy and possibility of application.Herein,the evaluated yield data of235U and239Pu are applied in the transport-burnup simulation of a pressurized water reactor(PWR)and sodium-cooled fast reactor(SFR)for verification.During the reactor operation stage,the errors in pin-cell reactivity caused by the evaluated fission yield do not exceed 500 and 200 pcm for the PWR and SFR,respectively.The errors in decay heat and135Xe and149Sm concentrations during the short-term shutdown of the PWR are all less than 1%;the errors in decay heat and activity of the spent fuel of the PWR and SFR during the temporary storage stage are all less than 2%.For the PWR,the errors in important nuclide concentrations in spent fuel,such as90Sr,137Cs,85Kr,and99Tc,are all less than 6%,and a larger error of 37%is observed on129I.For the SFR,the concentration errors of ten important nuclides in spent fuel are all less than 16%.A comparison of various aspects reveals that the transport-burnup simulation results using the FYTD model evaluation have little difference compared with the reference results using ENDF/B-Ⅷ.0 data.This proves that the evaluation of the FYTD model may have application value in reactor physical analysis.

Analysis of rockburst mechanism and warning based on microseismic moment tensors and dynamic Bayesian networks

One of the major factors inhibiting the construction of deep underground projects is the risk posed by rockbursts.A study was conducted on the access tunnel of the Shuangjiangkou hydropower station to determine the evolutionary mechanism of microfractures within the surrounding rock mass during rockburst development and develop a rockburst warning model.The study area was chosen through the combination of field studies with an analysis of the spatial and temporal distribution of microseismic(MS)events.The moment tensor inversion method was adopted to study rockburst mechanism,and a dynamic Bayesian network(DBN)was applied to investigating the sensitivity of MS source parameters for rockburst warnings.A MS multivariable rockburst warning model was proposed and validated using two case studies.The results indicate that fractures in the surrounding rock mass during the development of strain-structure rockbursts initially show shear failure and are then followed by tensile failure.The effectiveness of the DBN-based rockburst warning model was demonstrated using self-validation and K-fold cross-validation.Moment magnitude and source radius are the most sensitive factors based on an investigation of the influence on the parent and child nodes in the model,which can serve as important standards for rockburst warnings.The proposed rockburst warning model was found to be effective when applied to two actual projects.

GPU-acceleration 3D rotated-staggered-grid solutions to microseismic anisotropic wave equation with moment tensor implementation

To improve the accuracy of microseismic inversion,seismic anisotropy and moment tensor source should be carefully considered in the forward modelling stage.In this study,3D microseismic anisotropy wave forward modelling with a moment tensor source was proposed.The modelling was carried out based on a rotated-staggered-grid(RSG)scheme.In contrast to staggered-grids,the RSG scheme defines the velocity components and densities at the same grid,as do the stress components and elastic parameters.Therefore,the elastic moduli do not need to be interpolated.In addition,the detailed formulation and implementation of moment-tensor source loaded on the RSG was presented by equating the source to the stress increments.Meanwhile,the RSG-based 3D wave equation forward modelling was performed in parallel using compute unified device architecture(CUDA)programming on a graphics processing unit(GPU)to improve its efficiency.Numerical simulations including homogeneous and anisotropic models were carried out using the method proposed in this paper,and compared with other methods to prove the reliability of this method.Furthermore,the high efficiency of the proposed approach was evaluated.The results show that the computational efficiency of proposed method can be improved by about two orders of magnitude compared with traditional central processing unit(CPU)computing methods.It could not only help the analysis of microseismic full wavefield records,but also provide support for passive source inversion,including location and focal mechanism inversion,and velocities inversion.

An Alternative Way to Mapping Cone: The Algebraic Topology of the Pinched Tensor

In this research, we explore the properties and applications of the mapping cone and its variant, the pinched mapping cone. The mapping cone is a construction that arises naturally in algebraic topology and is used to study the homotopy type of spaces. It has several key properties, including its homotopy equivalence to the cofiber of a continuous map, and its ability to compute homotopy groups using the long exact sequence associated with the cofiber. We also provide an overview of the properties and applications of the mapping cone and the pinched mapping cone in algebraic topology. This work highlights the importance of these constructions in the study of homotopy theory and the calculation of homotopy groups. The study also points to the potential for further research in this area which includes the study of higher homotopy groups and the applications of these constructions to other areas of mathematics.

Research on infrared dim and small target detection algorithm based on low-rank tensor recovery

In order to rapidly and accurately detect infrared small and dim targets in the infrared image of complex scene collected by virtual prototyping of space-based downward-looking multiband detection,an improved detection algorithm of infrared small and dim target is proposed in this paper.Firstly,the original infrared images are changed into a new infrared patch tensor mode through data reconstruction.Then,the infrared small and dim target detection problems are converted to low-rank tensor recovery problems based on tensor nuclear norm in accordance with patch tensor characteristics,and inverse variance weighted entropy is defined for self-adaptive adjustment of sparseness.Finally,the low-rank tensor recovery problem with noise is solved by alternating the direction method to obtain the sparse target image,and the final small target is worked out by a simple partitioning algorithm.The test results in various spacebased downward-looking complex scenes show that such method can restrain complex background well by virtue of rapid arithmetic speed with high detection probability and low false alarm rate.It is a kind of infrared small and dim target detection method with good performance.

Application of tensor CSAMT with high-power orthogonal signal sources in Jiama porphyry copper deposit,South Tibet

The Jiama porphyry copper deposit in Tibet is one of the proven supergiant copper deposits in the Qinghai-Tibet Plateau at present,with the reserves of geological resources equivalent to nearly 20×10^(6) t.However,it features wavy and steep terrain,leading to extremely difficult field operation and heavy interference.This study attempts to determine the effects of the tensor controlled-source audiomagnetotellurics(CSAMT)with high-power orthogonal signal sources(also referred to as the high-power tensor CSAMT)when it is applied to the deep geophysical exploration in plateaus with complex terrain and mining areas with strong interference.The test results show that the high current provided by the highpower tensor CSAMT not only greatly improved the signal-to-noise ratio but also guaranteed that effective signals were received in the case of a long transmitter-receiver distance.Meanwhile,the tensor data better described the anisotropy of deep geologic bodies.In addition,the tests also show that when the transmitting current reaches 60 A,it is still guaranteed that strong enough signals can be received in the case of the transmitter-receiver distance of about 25 km,sounding curves show no near field effect,and effective exploration depth can reach 3 km.The 2D inversion results are roughly consistent with drilling results,indicating that the high-power tensor CSAMT can be used to achieve nearly actual characteristics of underground electrical structures.Therefore,this method has great potential for application in deep geophysical exploration in plateaus and mining areas with complex terrain and strong interference,respectively.This study not only serves as important guidance on the prospecting in the Qinghai-Tibet Plateau but also can be used as positive references for deep mineral exploration in other areas.

Separable Symmetric Tensors and Separable Anti-symmetric Tensors

In this paper,we first initialize the S-product of tensors to unify the outer product,contractive product,and the inner product of tensors.Then,we introduce the separable symmetry tensors and separable anti-symmetry tensors,which are defined,respectively,as the sum and the algebraic sum of rank-one tensors generated by the tensor product of some vectors.We offer a class of tensors to achieve the upper bound for rank(A)≤6 for all tensors of size 3×3×3.We also show that each 3×3×3 anti-symmetric tensor is separable.

Tensor Train Random Projection

This work proposes a Tensor Train Random Projection(TTRP)method for dimension reduction,where pairwise distances can be approximately preserved.Our TTRP is systematically constructed through a Tensor Train(TT)representation with TT-ranks equal to one.Based on the tensor train format,this random projection method can speed up the dimension reduction procedure for high-dimensional datasets and requires fewer storage costs with little loss in accuracy,comparedwith existingmethods.We provide a theoretical analysis of the bias and the variance of TTRP,which shows that this approach is an expected isometric projectionwith bounded variance,and we show that the scaling Rademacher variable is an optimal choice for generating the corresponding TT-cores.Detailed numerical experiments with synthetic datasets and theMNIST dataset are conducted to demonstrate the efficiency of TTRP.

Diffusion tensor imaging of horizontal extraocular muscles in patients with concomitant and paralytic esotropia

AIM:To assess metrics of diffusion tensor imagining(DTI)in evaluating microstructural abnormalities of horizontal extraocular muscles(EOM)in esotropia.METHODS:Six adult concomitant esotropia patients,5 unilateral abducent paralysis patients and 2 healthy volunteers were enrolled.Conventional magnetic resonance imaging(MRI)and DTI were performed on all subjects using 3T MR scanner.Fractional anisotropy(FA),mean diffusivity(MD),axial diffusivity(AD),and radial diffusivity(RD)of medial and lateral rectus muscles were measured and compared between patients group and control group.RESULTS:Medial rectus MD and RD within the adducted eye of concomitant patients was significantly greater than that in unilateral abducent paralysis patients(0.259×10-2 mm^(2)/s vs 0.207×10-2 mm^(2)/s,P=0.014;0.182×10-2 mm^(2)/s vs 0.152×10-2 mm^(2)/s,P=0.017).Both strabismus patients showed a significantly decreased MD and AD than that obtained in normal controls for lateral rectus muscles(P<0.05).Medial rectus MD of the adducted eye in concomitant strabismus patients was significantly decreased than that in healthy controls(0.259×10-2 mm^(2)/s vs 0.266×10-2 mm^(2)/s,P=0.010).Lateral rectus AD of the adducted eye in concomitant strabismus patients was significantly decreased as compared with that in healthy controls(0.515×10-2 mm^(2)/s vs 0.593×10-2 mm^(2)/s,P=0.013).No statistically significant differences were present between the adducted and fixating eyes in concomitant strabismus patients.CONCLUSION:DTI represents a feasible technique to assess tissue characteristics of EOM.The effects of eye position changes on DTI parameters are subtle.Decreased MD and RD could be evidence for remodeling of the medial rectus muscle contracture.Lower medial and lateral recuts MD of concomitant esotropia patients indicates a thinner fibrous structure of the EOM.Lower MD and AD should be general character of esotropia.

An Efficient Randomized Fixed-Precision Algorithm for Tensor Singular Value Decomposition

The existing randomized algorithms need an initial estimation of the tubal rank to compute a tensor singular value decomposition.This paper proposes a new randomized fixed-precision algorithm which for a given third-order tensor and a prescribed approximation error bound,it automatically finds the tubal rank and corresponding low tubal rank approximation.The algorithm is based on the random projection technique and equipped with the power iteration method for achieving better accuracy.We conduct simulations on synthetic and real-world datasets to show the efficiency and performance of the proposed algorithm.

A Vector Tensor Calculus Description of a Euclidean Space

We present a tensor description of Euclidean spaces that emphasizes the use of geometric vectors which leads to greater geometric insight and a higher degree of organization in analytical expressions. We demonstrate the effectiveness of the approach by proving a number of integral identities with vector integrands. The presented approach may be aptly described as absolute vector calculus or as vector tensor calculus.