Cyclic Solution and Optimal Approximation of the Quaternion Stein Equation

In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the target structure matrix is constructed by using the complex decomposition of the quaternion matrix, to obtain the necessary and sufficient conditions for the existence of the cyclic solution of the equation and the expression of the general solution. Secondly, the Stein equation is converted into the Sylvester equation by adding the necessary parameters, and the condition for the existence of a cyclic solution and the expression of the equation’s solution are then obtained by using the real decomposition of the quaternion matrix and the Kronecker product of the matrix. At the same time, under the condition that the solution set is non-empty, the optimal approximation solution to the given quaternion circulant matrix is obtained by using the property of Frobenius norm property. Numerical examples are given to verify the correctness of the theoretical results and the feasibility of the proposed method. .

Representation of Physical Fields as Einstein Manifold

In this work we investigate the possibility to represent physical fields as Einstein manifold. Based on the Einstein field equations in general relativity, we establish a general formulation for determining the metric tensor of the Einstein manifold that represents a physical field in terms of the energy-momentum tensor that characterises the physical field. As illustrations, we first apply the general formulation to represent the perfect fluid as Einstein manifold. However, from the established relation between the metric tensor and the energy-momentum tensor, we show that if the trace of the energy-momentum tensor associated with a physical field is equal to zero then the corresponding physical field cannot be represented as an Einstein manifold. This situation applies to the electromagnetic field since the trace of the energy-momentum of the electromagnetic field vanishes. Nevertheless, we show that a system that consists of the electromagnetic field and non-interacting charged particles can be represented as an Einstein manifold since the trace of the corresponding energy-momentum of the system no longer vanishes. As a further investigation, we show that it is also possible to represent physical fields as maximally symmetric spaces of constant scalar curvature.

Fast 3D forward modeling of the magnetic field and gradient tensor on an undulated surface

Magnetic field gradient tensor technique provides abundant data for delicate inversion of subsurface magnetic susceptibility distribution. Large scale magnetic data inversion imaging requires high speed and accuracy for forward modeling. For arbitrarily distributed susceptibility data on an undulated surface, we propose a fast 3D forward modeling method in the wavenumber domain based on(1) the wavenumber-domain expression of the prism combination model and the Gauss–FFT algorithm and(2) cubic spline interpolation. We apply the proposed 3D forward modeling method to synthetic data and use weighting coefficients in the wavenumber domain to improve the modeling for multiple observation surfaces, and also demonstrate the accuracy and efficiency of the proposed method.

Combination of structure tensor and tilt angle in the edge detection of potential field data

Edge detection is a commonly requested task in the interpretation of potential field data. Different methods have different results for varied depths and shapes of geological bodies. In this paper,we propose using the combination of structure tensor and tilt angle to detect the edges of the sources,which can display the edges of shallow and deep bodies simultaneously. Through tests on synthetic potential field data,it is obvious that the proposed edge detection methods can display the sources edges more clearly and precisely,compared with other commonly used methods. The application on real potential field data shows similar result,obtaining the edges of layers and faults clearly. In addition,another advantage of the new method is its insensitivity to noise.

Calculation of the Centrally Symmetric and Vortex Forces Acting on Momentum in Distortion Tensor Field: Explanation of High-Temperature Plasma and Vortex Motions

The paper deals with calculation of the centrally symmetric and vortex forces for the momentum of a particle in the distortion tensor field from the action minimum, by analogy with the calculation of forces for a charge in an electromagnetic field. It is demonstrated that: 1) The compensating interaction tensor corresponds to the distortion tensor in a solid. 2) The centrally symmetric force of the distortion tensor acts on the momentum as a charge, and is analogous to the Coulomb force. In a gas, it results in change in the momentum value of the molecules exponentially to some extent. The action of this force explains the high-temperature plasma in the gas. 3) The vortex force of the distortion tensor is equivalent to the Peach-Koehler force in a solid. It acts on the momentum flow, similar to the Lorentz magnetic force, and explains the vortex motions in space, in the form of “black holes”, and in the atmosphere, in the form of cyclones and anticyclones.

Spatio-temporal variation of the stress field in the Wenchuan aftershock region

Focal mechanism solutions and centroid depths of 312 M≥4 aftershocks from the 2008 Wenchuan earthquake sequence have been derived by CAP （Cut and Paste） method from broadband waveform data with relatively high signal-to-noise ratio （SNR）. Following this, we have analyzed the distribution of focal depths and the stress tensors, as well as the types of focal mechanisms. The major results are： （1） different cross-sections show that the depth ranges of the aftershocks at the southern and northern ends of the aftershock area along the Longmenshan fault zone are wider than those on the central segment, where rare M≥4 aftershocks occurred at depths shallower than 10 kin. The main faults trend to the NW on the southern and central segments, and for the northern segment, no dominant trend direction has been determined; （2） stress tensor distribution demonstrates that the majority of the aftershock areas on the cross-section along the major axis are mainly under compressive stress perpendicular to the profile; however, for the areas near Lixian, Beichuan, Qingchuan and the shallow parts of its northern segment, large principal stress components are parallel to the major axis profile direction. On the cross-sections perpendicular to the major axis, the three areas above can be divided into two parts： one with dominantly compressional stress near the major faults of the Longmenshan fault zone on the SE side, and the other with NE-direction push along the fault zone on the NW side; （3） the stress tensor distribution in map view is very similar to those on the vertical cross-sections. In map view, the orientation of the principal compressional stress axis $1 on the central segment of the aftershock area presents an SE-trending arc shape; （4） the stress tensor slices at different depths show that the orientation of S1 axis mainly changes on the central segment and at the northern end, indicating that the two segments have different seismogenic structures at different depths; （5） with the exception of the northern end of the aftershock region, the orientation of the $1 axis changes little during the early and late stages, illustrating the seismogenic structures are relatively stable; （6） preliminary analyses for the seismogenic structures at the northern end indicated that deeper strike-slip quakes occurred on the ENE-striking branch at first, and then the NNE-striking branch faults at the northern end were activated and generated a series of relatively shallow strike-slip earthquakes due to subsequent stress-triggering; （7） the aftershock triggering mechanism that occurred near Lixian is different between the shallow and deep depths, and between the early and late stages, indicating that the main faults and the branch faults responsible for aftershocks are at different depths. Consequently, the relaxation effect of the main shock particularIy impacts the branch faults.

Visualizing diffusion tensor fields on streamsurfaces with merging ellipsoids

Streamsurfaces in diffusion tensor fields are used to represent structures with pri- marily planar diffusion. So far, however, no effort has been made on the visualization of the anisotropy of diffusion on them, although this information is very important to identify the problematic regions of these structures. We propose two methods to display this anisotropy information. The first one employs a set of merging ellipsoids, which simultaneously character- ize the local tensor details - anisotropy - on them and portray the shape of the streamsurfaces. The weight between the streamsurfaces continuity and the discrete local tensors can be inter- actively adjusted by changing some given parameters. The second one generates a dense LIC （line integral convolution） texture of the two tangent eigenvector fields along the streamsurfaces firstly, and then blends in some color mapping indicating the anisotropy information. For high speed and high quality of texture images, we confine both the generation and the advection of the LIC texture in the image space. Merging ellipsoids method reveals the entire anisotropy information at discrete points by exploiting the geometric attribute of ellipsoids, and thus suits for local and detailed examination of the anisotropy; the texture-based method gives a global representation of the anisotropy on the whole streamsurfaces with texture and color attributes. To reveal the anisotropy information more efficiently, we integrate the two methods and use them at two different levels of details.

Plane Symmetric Solutions to the Nonlinear Spinor Field Equations in General Relativity Theory

We have obtained exact static plane-symmetric solutions to the spinor field equations with nonlinear terms which are arbitrary functions of invariant , taking into account their own gravitational field. It is shown that the initial set of the Einstein and spinor field equations with a power-law nonlinearity have regular solutions with a localized energy density of the spinor field only if m=0 (m is the mass parameter in the spinor field equations). Equations with power and polynomial nonlinearities are studied in detail. In this case, a soliton-like configuration has negative energy. We have also obtained exact static plane-symmetric solutions to the above spinor field equations in flat space-time. It is proved that in this case soliton-like solutions are absent.

Fingerprint singular points extraction based on orientation tensor field and Laurent series

Singular point(SP)extraction is a key component in automatic fingerprint identification system(AFIS).A new method was proposed for fingerprint singular points extraction,based on orientation tensor field and Laurent series.First,fingerprint orientation flow field was obtained,using the gradient of fingerprint image.With these gradients,fingerprint orientation tensor field was calculated.Then,candidate SPs were detected by the cross-correlation energy in multi-scale Gaussian space.The energy was calculated between fingerprint orientation tensor field and Laurent polynomial model.As a global descriptor,the Laurent polynomial coefficients were allowed for rotational invariance.Furthermore,a support vector machine(SVM)classifier was trained to remove spurious SPs,using cross-correlation coefficient as a feature vector.Finally,experiments were performed on Singular Point Detection Competition 2010(SPD2010)database.Compared to the winner algorithm of SPD2010 which has best accuracy of 31.90%,the accuracy of proposed algorithm is 45.34%.The results show that the proposed method outperforms the state-of-the-art detection algorithms by large margin,and the detection is invariant to rotational transformations.

Underwater square-root cubature attitude estimator by use of quaternion-vector switching and geomagnetic field tensor

This paper presents a kind of attitude estimation algorithm based on quaternion-vector switching and square-root cubature Kalman filter for autonomous underwater vehicle(AUV).The filter formulation is based on geomagnetic field tensor measurement dependent on the attitude and a gyro-based model for attitude propagation. In this algorithm, switching between the quaternion and the three-component vector is done by a couple of the mathematical transformations. Quaternion is chosen as the state variable of attitude in the kinematics equation to time update, while the mean value and covariance of the quaternion are computed by the three-component vector to avoid the normalization constraint of quaternion. The square-root forms enjoy a continuous and improved numerical stability because all the resulting covariance matrices are guaranteed to stay positively semidefinite. The entire square-root cubature attitude estimation algorithm with quaternion-vector switching for the nonlinear equality constraint of quaternion is given. The numerical simulation of simultaneous swing motions in the three directions is performed to compare with the three kinds of filters and the results indicate that the proposed filter provides lower attitude estimation errors than the other two kinds of filters and a good convergence rate.

Moment tensor inversion for focal mechanism of the Beibuwan earthquakes

Two earthquakes of Ms=6.0 and Ms=6. 1 consecutively occurred on December 31, 1994 and January 10, 1995 in Beibuwan region, China. By using the generalized reflection-transmission coefficient matrix and the discrete slowness integration method in the calculation of Green's functions, we obtained the focal mechanisms of these earthquakes using long-period waveforms of regional body waves recorded by the China Digital Seismograph Network (CDSN) by means of moment tensor inversion method in frequency domain. The results inverted indicate that the focal mechanisms of these two earthquakes were similar to each other. Their principal compressional stresses are in NW-SE direction and principal tensional stresses are in NE-SW direction. It turns out that the occurrence of the two earthquakes was controlled by the same tectonic environment related to the collision of the Philippine Plate and the Eurasian Plates. On the other hand, the results imply that the stress field in the seismogenic region has a significant change after the Ms=6.0 earthquake. It may be proposed that the occurrence of the Ms=6. 1 earthquake could be related to the stress field adjustment caused by the Ms=6.0 earthquake.

Nonlinear Spinor Field Equations in Gravitational Theory: Spherical Symmetric Soliton-Like Solutions

This paper deals with an extension of a previous work [Gravitation & Cosmology, Vol. 4, 1998, pp 107-113] to exact spherical symmetric solutions to the spinor field equations with nonlinear terms which are arbitrary functions of S=ψψ, taking into account their own gravitational field. Equations with power and polynomial nonlinearities are studied in detail. It is shown that the initial set of the Einstein and spinor field equations with a power nonlinearity has regular solutions with spinor field localized energy and charge densities. The total energy and charge are finite. Besides, exact solutions, including soliton-like solutions, to the spinor field equations are also obtained in flat space-time.

How to Determine a Jump in Energy Prior to a Causal Barrier, with an Attendant Current, for an Effective Initial Magnetic Field. In the Pre Planckian to Planckian Space-Time

We start where we use an inflaton value due to use of a scale factor . Also we use as the variation of the time component of the metric tensor in Pre-Planckian space-time. Our objective is to find an effective magnetic field, to obtain the minimum scale factor in line with Non Linear Electrodynamics as given by Camara, et al., 2004. Our suggestion is based upon a new procedure for an effective current based upon an inflaton time exp (i times (frequency) times (cosmological time)) factor as a new rescaled inflaton which is then placed right into a Noether Current scalar field expression as given by Peskins, 1995. This is before the Causal surface with which is, right next to a quantum bounce, determined by , with the next shift in the Hubble parameter as set up to be then . And is an initial degree of freedom value of about 110. Upon calculation of the current, and a resulting magnetic field, for the space time bubble, we then next obtain a shift in energy, leading to a transition from too. We argue then that the delineation of the term is a precursor to filling in information as to the Weyl Tensor for near singularity measurements of starting space-time. Furthermore, as evidenced in Equations ((26) and (27)) of this document, we focus upon a “first order” that checks into if a cosmological “constant” would be invariant in time, or would be along the trajectory of the time, varying Quinessence models. We close this document, with Maxwell equations as to Post Newtonian theory, for Gravity, with our candidates as to a magnetic field included in, with what we think this pertains to, as far as Gravo Electric and Gravo Magnetic fields, and then make suggestions as to a quantum version of this methodology for future gravitational wave physics research. This is Appendix G, this last topic, and deliberately set up future works paradigm which will be investigated in the coming year. It is based upon a Gravo Electric potential, and we make suggestions as to its upgrade in our future works, in early universe cosmology. In the reference by Poisson, and Will, they write and in this last section we come up with a value of U, based in part on the comparison with the alteration of velocity, due to a massive graviton, namely via the substitution, we write as , so as to come up with a post Newtonian approximation result for a magnetic field. We compare this magnetic field, as far as the Inflaton magnetic field, and use it to come up with observations with regards to the phenomenology of gravity in Pre Planckian to Planckian regime limits. We close, then with the observation given in Appendix H, of the inhomogeneity of Pre Planckian-to Planckian space time as a necessary condition for a Gravi-Magnetic field. We also reference an Appendix I, which does a summary of a 5th force calculation, and we then compare those results, with our temporary results of a Gravi Magneitc field, as we have tried to start up as a future works project.

On the Field Equations of General Relativity

In this work, we examine the geometric character of the field equations of general relativity and propose to formulate relativistic field equations in terms of the Riemann curvature tensor. The resulted relativistic field equations are also integrated into the general framework that we have presented in our previous works that all known classical fields can be expressed in the same dynamical form. We also discuss a possibility to reformulate the field equations of general relativity so that the Ricci curvature tensor and the energy-momentum tensor can appear symmetrically in the field equations without violating the conservation law stated by the covariant derivative.

A Locally Conservative Energy-Momentum Tensor in the General Relativity Based on a Cosmological Model without Singularity

According to the conventional theory it is difficult to define the energy-momentum tensor which is locally conservative. The energy-momentum tensor of the gravitational field is defined. Based on a cosmological model without singularity, the total energy-momentum tensor is defined which is locally conservative in the general relativity. The tensor of the gravitational mass is different from the energy-momentum tensor, and it satisfies the gravitational field equation and its covariant derivative is zero.

Moment tensor inversion of focal mechanism for the aftershock sequence of 1982 Lulong M_S =6.1 earthquake

Using the moment tensor inversion method, we calculate the focal mechanisms of the aftershock sequence of the Ms=6.1 Lulong earthquake occurred on October 19, 1982 in Hebei Province. We found that the pressure axis in Lulong basin is nearly in the east-west direction with an azimuth of N74°E. However, in the north of the basin the stress axis changes to N43°E; and in some places near the center of the basin it changes to the northwest that is almost perpendicular to the P axis obtained by us from those events around the basin. This feature illuminates that in Lulong earthquake sequence, the stress direction is different in different parts of crustal structure, which shows that the tectonic movement in Lulong region is complex. This is because that Lulong is located in the eastern part of Chinese mainland and is subject to the compression of Japanese Sea Basin driven by the Pacific Plate. On the other hand, nipped by the Yanshan and North China blocks, Lulong is obviously restricted by the block boundaries.

Determinations of directions of the mean stress field in Sichuan-Yunnan region from a number of focal mechanism solutions

Based on the spatial orientation and slip direction of the fault plane solutions, we present the expression of corresponding mechanical axis tensor in geographic coordinate system, and then put forward a method for calculating average mechanical axis tensor and its eigenvalues, which involves solving the corresponding eigenequation. The method for deducing mean stress field from T, B, and P axes parameters of a number of focal mechanism solutions has been verified by inverting data of mean stress fields in Fuyun region and in Tangshan region with fitting method of slip direction, and both results are consistent. To study regional average stress field, we need to choose a population of focal mechanism solutions of earthquakes in the massifs where there are significant tectonic structures. According to the focal mechanism solutions of 256 moderate-strong earthquakes occurred in 13 seismic zones of Sichuan-Yunnan region, the quantitative analysis results of stress tensor in each seismic zone have been given. The algorithm of such method is simple and convenient, which makes the method for analyzing tectonic stress field with large amount of focal mechanism solution data become quantified.

Tensor Charge of the Nucleon from QCD Sum Rules

The nucleon's tensor charges(isovector g_(T)^(v)=δu-δd and isoscalar g_(T)^(s)=δu+δd)are calculated using the QCD(quantum chromodynamics)sum rules in the presence of an external tensor field.By combining the mass sum rule of nucleon,we derive a simplified form for proton tensor charges,which gives g_(T)^(v)=1.27 and g_(T)^(s)=1.31,or δu=1.29 and δd=0.02 at the scale of nucleon mass,with an uncertainty about 20%.

Note on Energy-Momentum Tensor for General Mixed Tensor-Spinor Fields

This note provides an explicit proof of the equivalence of Belinfante's energy-momentum tensor and metric energy-momentum tensor for general mixed tensor-spinor fields.

On Quasi-Einstein Field Equation

In this paper some properties of a symmetric tensor field T（X,Y） = g（A（X）, Y） on a Riemannian manifold （M, g） without boundary which satisfies the S quasi-Einstein equation Rij-S/2gij=Tij＋bξiξj are given. The necessary and sufficient conditions for this tensor to satisfy the quasi-Einstein equation are also obtained.