Second Approximation of the Generalized Planetary Equation Based upon Golden Metric Tensors

In this paper, we consider the Post Einstein Planetary equation of motion. We succeeded in offering a solution using second approximation method, in which we obtained eight exact mathematical solutions that rebel amazing theoretical results. To the order of C-2, two of these exact solutions are reduced to the approximate solutions from the method of successive approximations.

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.

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.

Positive definiteness of fourth-order partially symmetric tensors

In this paper,we consider the positive definiteness of fourth-order partially symmetric tensors.First,two analytically sufficient and necessary conditions of positive definiteness are provided for fourth-order two dimensional partially symmetric tensors.Then,we obtain several sufficient conditions for rank-one positive definiteness of fourth-order three dimensional partially symmetric tensors.

Full gravity gradient tensors from vertical gravity by cosine transform

We present a method to calculate the full gravity gradient tensors from pre-existing vertical gravity data using the cosine transform technique and discuss the calculated tensor accuracy when the gravity anomalies are contaminated by noise. Gravity gradient tensors computation on 2D infinite horizontal cylinder and 3D ＂Y＂ type dyke models show that the results computed with the DCT technique are more accurate than the FFT technique regardless if the gravity anomalies are contaminated by noise or not. The DCT precision has increased 2 to 3 times from the standard deviation. In application, the gravity gradient tensors of the Hulin basin calculated by DCT and FFT show that the two results are consistent with each other. However, the DCT results are smoother than results computed with FFT. This shows that the proposed method is less affected by noise and can better reflect the fault distribution.

Edge enhancement of gravity anomalies and gravity gradient tensors using an improved small sub-domain filtering method

In order to enhance geological body boundary visual effects in images and improve interpretation accuracy using gravity and magnetic field data, we propose an improved small sub-domain filtering method to enhance gravity anomalies and gravity gradient tensors. We discuss the effect of Gaussian white noise on the improved small sub-domain filtering method, as well as analyze the effect of window size on geological body edge recognition at different extension directions. Model experiments show that the improved small sub-domain filtering method is less affected by noise, filter window size, and geological body edge direction so it can more accurately depict geological body edges than the conventional small sub-domain filtering method. It also shows that deeply buried body edges can be well delineated through increasing the filter window size. In application, the enhanced gravity anomalies and calculated gravity gradient tensors of the Hulin basin show that the improved small sub-domain filtering can recognize more horizontal fault locations than the conventional method.

A PID-incorporated Latent Factorization of Tensors Approach to Dynamically Weighted Directed Network Analysis

A large-scale dynamically weighted directed network(DWDN)involving numerous entities and massive dynamic interaction is an essential data source in many big-data-related applications,like in a terminal interaction pattern analysis system(TIPAS).It can be represented by a high-dimensional and incomplete(HDI)tensor whose entries are mostly unknown.Yet such an HDI tensor contains a wealth knowledge regarding various desired patterns like potential links in a DWDN.A latent factorization-of-tensors(LFT)model proves to be highly efficient in extracting such knowledge from an HDI tensor,which is commonly achieved via a stochastic gradient descent(SGD)solver.However,an SGD-based LFT model suffers from slow convergence that impairs its efficiency on large-scale DWDNs.To address this issue,this work proposes a proportional-integralderivative(PID)-incorporated LFT model.It constructs an adjusted instance error based on the PID control principle,and then substitutes it into an SGD solver to improve the convergence rate.Empirical studies on two DWDNs generated by a real TIPAS show that compared with state-of-the-art models,the proposed model achieves significant efficiency gain as well as highly competitive prediction accuracy when handling the task of missing link prediction for a given DWDN.

Random quadralinear forms and schur product on tensors

In this work, we made progress on the problem that rpqlll哪(( is a Banach algebra under schur product. Our results extend Tonges results. We also obtained estimates for the norm of the random quadralinear form A: MNKHrpqsllll创串C, defined by: A(ei, ej, ek, es)=ijksa, where the (aijks)s are uniformly bounded, independent, mean zero random variables. We proved that under some conditions rpqsllll哪?(( is not a Banach algebra under schur product.

Preservation of Linear Constraints in Approximation of Tensors

For an arbitrary tensor(multi-index array) with linear constraints at each direction,it is proved that the factors of any minimal canonical tensor approximation to this tensor satisfy the same linear constraints for the corresponding directions.

Complex Spacetime Frame: Four-Vector Identities and Tensors

This paper provides derivation of some basic identities for complex four-component vectors defined in a complex four-dimensional spacetime frame specified by an imaginary temporal axis. The resulting four-vector identities take exactly the same forms of the standard vector identities established in the familiar three-dimensional space, thereby confirming the consistency of the definition of the complex four-vectors and their mathematical operations in the general complex spacetime frame. Contravariant and covariant forms have been defined, providing appropriate definitions of complex tensors, which point to the possibility of reformulating differential geometry within a spacetime frame.

Partial Quantum Tensors of Input and Output Connections

I show how many connections of Γ?are presently existing from R?to β?as they are being inputted simultaneously through tensor products. I plan to address the Quantum state of this tensor connection step by step throughout the application presented. Also, I will show you how to prove that the connection is true for this tensor connection through its output method using a small bit of tensor calculus and mostly number theory.

On Kirichenko Tensors of Nearly-Khlerian Manifolds

A short description of structural and virtual Kirichenko tensors that form a complete system of first-order differential-geometrical invariants of an arbitrary almost Hermitian structure is given.A characterization of nearly-Khlerian structures in terms of Kirichenko tensors is also given.

CANONICAL REPRESENTATIONS AND DEGREE OF FREEDOM FORMULAE OF ORTHOGONAL TENSORS IN N-DIMENSIONAL EUCLIDEAN SPACE

In this paper, with the help of the eigenvalue properties of orthogonal tensors in n-dimensional Euclidean space and the representations of the orthogonal tensors in 2-dimensional space, the canonical representations of orthogonal tensors in n-dimensional Euclidean space are easily gotten. The paper also gives all the constraint relationships among the principal invariants of arbitrarily given orthogonal tensor by use of Cayley-Hamilton theorem; these results make it possible to solve all the eigenvalues of any orthogonal tensor based on a quite reduced equation of m-th order, where m is the integer part ofn \2. Finally, the formulae of the degree of freedom of orthogonal tensors are given.

RESEARCH ON THE DEFORMATIONS IN QINGHAI_TIBET PLATEAU AND ITS MARGINS BY INVERTING SEISMIC MOMENT TENSORS AND GPS VELOCITIES

We have determined approximate average rates of deformation in the Qinghai_Tibet plateau and its margins from the GPS data for last 10 years and the moment tensors from earthquakes between 1900 and 1999.We also determined the strain rate (seismic strain rate) associated with the seismic deformation using 254 M w ≥5.0 earthquakes,and estimated the shortening and extension rates for every block in the area as well.We also estimated the strain rate (geodetic strain rate)by 80 GPS sites’ velocity vectors and analyzed characteristic of kinematics by two kinds of strain rates and discussed earthquake potential in the area.As a result,the deformation rates from seismic moment tensors and from GPS velocities are basically agreed with each other.It is feasible to analyze seismic risk by comparing geodetic strain rate with seismic strain rate based on the opinion that strain energy will be released through earthquake.It is concluded that there is no strong earthquake potential (>M7) in the Qinghai_Tibet plateau and its margins,but there is earthquake potential (>M5) in middle Tibet in a few years.

Weak WT_2-class of differential forms and weakly A-harmonic tensors

In this paper, we give the definition of weak WT2-class of differential forms, and then obtain its weak reverse Holder inequality. As an application, we give an alternative proof of the higher integrability result of weakly A-harmonic tensors due to B. Stroffolini.

Second Order Parallel Tensors on Quasi-constant Curvature Manifolds

The author establishes in this paper the following results: (1) In a quasiconstant curvature manifold M a parallel tensor of type is constant multiple of the metric tensor. (2) On a quasi_constant curvature manifold there is no nonzero parallel 2_form. Unless the Ricci principal curvature corresponding to the generator of M is equal to zero.

An irreducible polynomial functional basis of two-dimensional Eshelby tensors

The two-dimensional (2D) Eshelby tensors are discussed. Based upon the complex variable method, an integrity basis of ten isotropic invariants of the 2D Eshelby tensors is obtained. Since an integrity basis is always a polynomial functional basis, these ten isotropic invariants are further proven to form an irreducible polynomial functional basis of the 2D Eshelby tensors.

Recovery of Corrupted Low-Rank Tensors

This paper studies the problem of recovering low-rank tensors, and the tensors are corrupted by both impulse and Gaussian noise. The problem is well accomplished by integrating the tensor nuclear norm and the l1-norm in a unified convex relaxation framework. The nuclear norm is adopted to explore the low-rank components and the l1-norm is used to exploit the impulse noise. Then, this optimization problem is solved by some augmented-Lagrangian-based algorithms. Some preliminary numerical experiments verify that the proposed method can well recover the corrupted low-rank tensors.

ORIENTATION DISTRIBUTION FUNCTIONS FOR MICROSTRUCTURES OF HETEROGENEOUS MATERIALS (Ⅱ)──CRYSTAL DISTRIBUTION FUNCTIONS AND IRREDUCIBLE TENSORS RESTRICTED BY VARIOUS MATERIAL SYMMETRIES

The explicit representations for tensorial Fourier expansion of 3_D crystal orientation distribution functions (CODFs) are established. In comparison with that the coefficients in the mth term of the Fourier expansion of a 3_D ODF make up just a single irreducible mth_order tensor, the coefficients in the mth term of the Fourier expansion of a 3_D CODF constitute generally so many as 2m+1 irreducible mth_order tensors. Therefore, the restricted forms of tensorial Fourier expansions of 3_D CODFs imposed by various micro_ and macro_scopic symmetries are further established, and it is shown that in most cases of symmetry the restricted forms of tensorial Fourier expansions of 3_D CODFs contain remarkably reduced numbers of mth_order irreducible tensors than the number 2m+1 . These results are based on the restricted forms of irreducible tensors imposed by various point_group symmetries, which are also thoroughly investigated in the present part in both 2_ and 3_D spaces.

ORIENTATION DISTRIBUTION FUNCTIONS FOR MICROSTRUCTURES OF HETEROGENEOUS MATERIALS (Ⅰ) ──DIRECTIONAL DISTRIBUTION FUNCTIONS AND IRREDUCIBLE TENSORS

In this two_part paper, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficients for orientation distribution functions (ODFs) and crystal orientation distribution functions (CODFs), which are scalar functions defined on the unit sphere and the rotation group, respectively. Recently it has been becoming clearer and clearer that concepts of ODF and CODF play a dominant role in various micromechanically_based approaches to mechanical and physical properties of heterogeneous materials. The theory of group representations shows that a square integrable ODF can be expanded as an absolutely convergent Fourier series of spherical harmonics and these spherical harmonics can further be expressed in terms of irreducible tensors. The fundamental importance of such irreducible tensorial coefficients is that they characterize the macroscopic or overall effect of the orientation distribution of the size, shape, phase, position of the material constitutions and defects. In Part (Ⅰ), the investigation about the irreducible tensorial Fourier expansions of ODFs defined on the N_dimensional (N_D) unit sphere is carried out. Attention is particularly paid to constructing simple expressions for 2_ and 3_D irreducible tensors of any orders in accordance with the convenience of arriving at their restricted forms imposed by various point_group (the synonym of subgroup of the full orthogonal group) symmetries. In the continued work -Part (Ⅱ), the explicit expression for the irreducible tensorial expansions of CODFs is established. The restricted forms of irreducible tensors and irreducible tensorial Fourier expansions of ODFs and CODFs imposed by various point_group symmetries are derived.