Search Results (17)

Accurately identifying coherent vortex structures (CVSs) in backward-facing step (BFS) flows remains a challenge, particularly in reconciling visual streamlines with mathematical criteria. In this study, high-resolution velocity fields were captured using particle image velocimetry (PIV) in a pressurized BFS setup. Instantaneous streamlines reveal distinct spiral patterns, vortex centers, and saddle points, consistent with physical definitions of vortices and offering intuitive guidance for CVS detection. However, conventional vortex identification methods often fail to reproduce these visual features. To address this, an improved Q-criterion method is proposed, based on the normalization of the velocity gradient tensor. This approach enhances the rotational contribution while suppressing shear effects, leading to improved agreement in vortex position and shape with those observed in streamlines. While the normalization process alters the representation of physical vortex strength, the method bridges qualitative visualization and quantitative analysis. This streamline-consistent identification framework facilitates robust CVS detection in separated flows and supports further investigations in vortex dynamics and turbulence control. Full article

The covariance tensors in statistics and Riemann curvature tensor in relativity theory are both biquadratic tensors that are weakly symmetric, but not symmetric in general. Motivated by this, in this paper, we consider nonsymmetric biquadratic tensors and extend M-eigenvalues to nonsymmetric biquadratic tensors by symmetrizing these tensors. We present a Gershgorin-type theorem for biquadratic tensors, and show that (strictly) diagonally dominated biquadratic tensors are positive semi-definite (definite). We introduce Z-biquadratic tensors, M-biquadratic tensors, strong M-biquadratic tensors, ${\mathrm{B}}_0$-biquadratic tensors, and B-biquadratic tensors. We show that M-biquadratic tensors and symmetric ${\mathrm{B}}_0$-biquadratic tensors are positive semi-definite, and that strong M-biquadratic tensors and symmetric B-biquadratic tensors are positive definite. A Riemannian Limited-memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) method for computing the smallest M-eigenvalue of a general biquadratic tensor is presented. Numerical results are reported. Full article

In this paper, we show that even-order Pascal tensors are positive-definite, and odd-order Pascal tensors are strongly completely positive. The significance of these is that our induction proof method also holds for some other families of completely positive tensors, whose construction satisfies certain rules, such that the inherence property holds. We show that for all tensors in such a family, even-order tensors would be positive-definite, and odd-order tensors would be strongly completely positive, as long as the matrices in this family are positive-definite. In particular, we show that even-order generalized Pascal tensors would be positive-definite, and odd-order generalized Pascal tensors would be strongly completely positive, as long as generalized Pascal matrices are positive-definite. We also investigate even-order positive-definiteness and odd-order strong complete positivity for fractional Hadamard power tensors. Furthermore, we study determinants of Pascal tensors. We prove that the determinant of the mth-order two-dimensional symmetric Pascal tensor is equal to the mth power of the factorial of $m-1$. Full article

In many robotic applications, creating a map is crucial, and 3D maps provide a method for estimating the positions of other objects or obstacles. Most of the previous research processes 3D point clouds through projection-based or voxel-based models, but both approaches have certain limitations. This paper proposes a hybrid localization and mapping method using stereo vision and LiDAR. Unlike the traditional single-sensor systems, we construct a pose optimization model by matching ground information between LiDAR maps and visual images. We use stereo vision to extract ground information and fuse it with LiDAR tensor voting data to establish coplanarity constraints. Pose optimization is achieved through a graph-based optimization algorithm and a local window optimization method. The proposed method is evaluated using the KITTI dataset and compared against the ORB-SLAM3, F-LOAM, LOAM, and LeGO-LOAM methods. Additionally, we generate 3D point cloud maps for the corresponding sequences and high-definition point cloud maps of the streets in sequence 00. The experimental results demonstrate significant improvements in trajectory accuracy and robustness, enabling the construction of clear, dense 3D maps. Full article

A weak f-contact structure, introduced in our recent works, generalizes the classical f-contact structure on a smooth manifold, and its characteristic distribution defines a totally geodesic foliation with flat leaves. We find the splitting tensor of this foliation and use it to show positive definiteness of the Jacobi operators in the characteristic directions and to obtain a topological obstruction (including the Adams number) to the existence of weak f-K-contact manifolds, and prove integral formulas for a compact weak f-contact manifold. Based on applications of the weak f-contact structure in Riemannian contact geometry considered in the article, we expect that this structure will also be fruitful in theoretical physics, e.g., in QFT. Full article

Our research focuses on the tangent space of a point on a four-dimensional Riemannian manifold. Besides having a positive definite metric, the manifold is endowed with an additional tensor structure of type $(1,1)$, whose fourth power is minus the identity. The additional structure is skew-circulant and compatible with the metric, such that an isometry is induced in every tangent space on the manifold. Both structures define an indefinite metric. With the help of the indefinite metric, we determine circles in different two-planes in the tangent space on the manifold. We also calculate the length and area of the circles. On a smooth closed curve, such as a circle, we define a vector force field. Further, we obtain the circulation of the vector force field along the curve, as well as the flux of the curl of this vector force field across the curve. Finally, we find a relation between these two values, which is an analog of the well-known Green’s formula in the Euclidean space. Full article

A model of spacetime is presented. It has an extension to five dimensions, and in five dimensions the geometry is the dual of the Euclidean geometry w.r.t. an arbitrary positive-definite metric. Dually flat geometries are well-known in the context of information geometry. The present work explores their role in describing the geometry of spacetime. It is shown that the positive-definite metric with its flat 5-d connection can coexist with a pseudometric for which the connection is that of Levi–Civita. The 4-d geodesics are characterized by five conserved quantities, one of which can be chosen freely and is taken equal to zero in the present work. An explicit expression for the parallel transport operators is obtained. It is used to construct a pseudometric for spacetime by choosing an arbitrary possibly degenerate inner product in the tangent space of a reference point, for instance, that of Minkowski. By parallel transport, one obtains a pseudometric for spacetime, the metric connection of which extends to a 5-d connection with vanishing curvature tensor. The de Sitter space is considered as an example. Full article

In this work, we perform a phenomenological derivation of the first- and second-order relativistic hydrodynamics of dissipative fluids. To set the stage, we start with a review of the ideal relativistic hydrodynamics from energy–momentum and particle number conservation equations. We then go on to discuss the matching conditions to local thermodynamical equilibrium, symmetries of the energy–momentum tensor, decomposition of dissipative processes according to their Lorentz structure, and, finally, the definition of the fluid velocity in the Landau and Eckart frames. With this preparatory work, we first formulate the first-order (Navier–Stokes) relativistic hydrodynamics from the entropy flow equation, keeping only the first-order gradients of thermodynamical forces. A generalized form of diffusion terms is found with a matrix of diffusion coefficients describing the relative diffusion between various flavors. The procedure of finding the dissipative terms is then extended to the second order to obtain the most general form of dissipative function for multiflavor systems up to the second order in dissipative fluxes. The dissipative function now includes in addition to the usual second-order transport coefficients of Israel–Stewart theory also second-order diffusion between different flavors. The relaxation-type equations of second-order hydrodynamics are found from the requirement of positivity of the dissipation function, which features the finite relaxation times of various dissipative processes that guarantee the causality and stability of the fluid dynamics. These equations contain a complete set of nonlinear terms in the thermodynamic gradients and dissipative fluxes arising from the entropy current, which are not present in the conventional Israel–Stewart theory. Full article

Positive definite homogeneous multivariate forms play an important role in polynomial problems and medical imaging, and the definiteness of forms can be tested using structured tensors. In this paper, we state the equivalence between the positive definite multivariate forms and the corresponding tensors, and explain the connection between the positive definite tensors with $\mathcal{H}$-tensors. Then, based on the notion of diagonally dominant tensors, some criteria for $\mathcal{H}$-tensors are presented. Meanwhile, with these links, we provide an iterative algorithm to test the positive definiteness of multivariate homogeneous forms and prove its validity theoretically. The advantages of the obtained results are illustrated by some numerical examples. Full article

When conducting numerical upscaling, either for a fractured or a porous medium, it is important to account for anisotropy because in general, the resulting upscaled conductivity is anisotropic. Measurements made at different scales also demonstrate the existence of anisotropy of hydraulic conductivity. At the “microscopic” scale, the anisotropy results from the preferential flatness of grains, presence of shale, or variation of grain size in successive laminations. At a larger scale, the anisotropy results from preferential orientation of highly conductive geological features (channels, fracture families) or alternations of high and low conductive features (stratification, bedding, crossbedding). Previous surveys of homogenization techniques demonstrate that a wide variety of approaches exists to define and calculate the equivalent conductivity tensor. Consequently, the resulting equivalent conductivities obtained by these different methods are not necessarily equal, and they do not have the same mathematical properties (some are symmetric, others are not, for example). We present an overview of different techniques allowing a quantitative evaluation of the anisotropic equivalent conductivity for heterogeneous porous media, via numerical simulations and, in some cases, analytical approaches. New approaches to equivalent permeability are proposed for heterogeneous media, as well as discontinuous (composite) media, and also some extensions to 2D fractured networks. One of the main focuses of the paper is to explore the relations between these various definitions and the resulting properties of the anisotropic equivalent conductivity, such as tensorial or non-tensorial behavior of the anisotropic conductivity; symmetry and positiveness of the conductivity tensor (or not); dual conductivity/resistivity tensors; continuity and robustness of equivalent conductivity with respect to domain geometry and boundary conditions. In this paper, we emphasize some of the implications of the different approaches for the resulting equivalent permeabilities. Full article

Identifying the positive definiteness of even-order real symmetric tensors is an important component in tensor analysis. $\mathcal{H}$-tensors have been utilized in identifying the positive definiteness of this kind of tensor. Some new practical criteria for identifying $\mathcal{H}$-tensors are given in the literature. As an application, several sufficient conditions of the positive definiteness for an even-order real symmetric tensor were obtained. Numerical examples are given to illustrate the effectiveness of the proposed method. Full article

A diffusion tensor models the covariance of the Brownian motion of water at a voxel and is required to be symmetric and positive semi-definite. Therefore, image processing approaches, designed for linear entities, are not effective for diffusion tensor data manipulation, and the existence of artefacts in diffusion tensor imaging acquisition makes diffusion tensor data segmentation even more challenging. In this study, we develop a spatial fuzzy c-means clustering method for diffusion tensor data that effectively segments diffusion tensor images by accounting for the noise, partial voluming, magnetic field inhomogeneity, and other imaging artefacts. To retain the symmetry and positive semi-definiteness of diffusion tensors, the log and root Euclidean metrics are used to estimate the mean diffusion tensor for each cluster. The method exploits spatial contextual information and provides uncertainty information in segmentation decisions by calculating the membership values for assigning a diffusion tensor at one voxel to different clusters. A regularisation model that allows the user to integrate their prior knowledge into the segmentation scheme or to highlight and segment local structures is also proposed. Experiments on simulated images and real brain datasets from healthy and Spinocerebellar ataxia 2 subjects showed that the new method was more effective than conventional segmentation methods. Full article

The positive definiteness of even-order weakly symmetric tensors plays important roles in asymptotic stability of time-invariant polynomial systems. In this paper, we establish two Brauer-type Z-eigenvalue inclusion sets with parameters by Z-identity tensors, and show that these inclusion sets are sharper than existing results. Based on the new Z-eigenvalue inclusion sets, we propose some sufficient conditions for testing the positive definiteness of even-order weakly symmetric tensors, as well as the asymptotic stability of time-invariant polynomial systems. The given numerical experiments are reported to show the efficiency of our results. Full article

We investigate whether Szabo’s metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern–Rund connection defines an affine connection on the underlying manifold), then it is affinely equivalent to a Riemann space, meaning that its affine connection is the Levi–Civita connection of some Riemannian metric. We show for the first time that this result does not extend to general Finsler spacetimes. More precisely, we find a large class of Berwald spacetimes for which the Ricci tensor of the affine connection is not symmetric. The fundamental difference from positive definite Finsler spaces that makes such an asymmetry possible is the fact that generally, Finsler spacetimes satisfy certain smoothness properties only on a proper conic subset of the slit tangent bundle. Indeed, we prove that when the Finsler Lagrangian is smooth on the entire slit tangent bundle, the Ricci tensor must necessarily be symmetric. For large classes of Finsler spacetimes, however, the Berwald property does not imply that the affine structure is equivalent to the affine structure of a pseudo-Riemannian metric. Instead, the affine structure is that of a metric-affine geometry with vanishing torsion. Full article

Surgery is the only definitive treatment for degenerative cervical myelopathy (DCM), however, the degree of neurological recovery is often unpredictable. Here, we assess the utility of a multidimensional diagnostic approach, consisting of clinical, neurophysiological, and radiological parameters, to identify patients likely to benefit most from surgery. Thirty-six consecutive patients were prospectively analyzed using the modified Japanese Orthopedic Association (mJOA) score, MEPs/SSEPs and advance and conventional MRI parameters, at baseline, and 3- and 12-month postoperatively. Patients were subdivided into “normal” and “best” responders (<50%, ≥50% improvement in mJOA), and correlation between Diffusion Tensor Imaging (DTI) parameters, mJOA, and MEP/SSEP latencies were examined. Twenty patients were “best” responders and 16 were “normal responders”, but there were no statistical differences in age, T2 hyperintensity, and midsagittal diameter between them. There was a significant inverse correlation between the MEPs central conduction time and mJOA in the preoperative period (p = 0.0004), and a positive correlation between fractional anisotropy (FA) and mJOA during all the phases of the study, and statistically significant at 1-year (r = 0.66, p = 0.0005). FA was significantly higher amongst “best responders” compared to “normal responders” preoperatively and at 1-year (p = 0.02 and p = 0.009). A preoperative FA > 0.55 was predictor of a better postoperative outcome. Overall, these results support the concept of a multidisciplinary approach in the assessment and management of DCM. Full article