Proper characterization of fracture reservoir is crucial for their sound development plan. It is however very difficult to correctly obtain various fracture reservoir properties such as Permeability due to high order of heterogeneity and anisotropy within these reservoirs. Classical dual porosity and/or dual Permeability models consider a regular fracture network across the reservoir. To improve the concept, we develop a numerical method for tonsorial Permeability calculation of blocks with random/disordered fracture distribution. We considered a 2D Cartesian fine grid in which the fractures were defined explicitly with their endpoints coordinates. Applying proper boundary conditions, single phase flow is then solved. Full Tensor Permeability is then obtained analytically from the calculated flow and pressure fields. The result of our method is compared well with that of the analytical models for simple fracture systems. In addition we reported the Permeability Tensor values of random fracture networks where no analytical solution is available.

The hydraulic properties of the rock masses are of great importance in analyzing the behavior and stability of the structures constructed on or in rock mass. Permeability is key parameter among other rock mass features due to its important role in rock mass overall behavior. According to aforementioned reason, numerous efforts have been made by researchers in the field of rock mechanics for its obtaining. To access the rock masses’ Permeability, in-situ test methods and simulation techniques could be used. In-situ tests like Lugeon Test are time-consuming and costly and they provide local results. Simulation base methods calculate the Permeability of the model that is generated similar to the real region indeed and the developing the results to the real condition always raises substantial challenges. according to the aforementioned reason, direct acquiring of Permeability with optimum cost and time which is easily generalizable to the overall of a region would be very important. In this work using crack Tensor concept, Permeability Tensor of Lorestan’s Rudbar dam cavern is calculated efficiently by considering rock mass structural features. Resulted Permeability of the power plant’s cavern was obtained equal to  that seems to be acceptable compared to the measured values which is obtained  9/87×10-7 m/s.

Multilinear Discriminant Analysis (MDA) is a powerful dimension reduction method specifically formulated to deal with Tensor data. Precisely, the goal of MDA  is to find mode-specific projections that optimally separate Tensor data from different classes. However, to solve this task, standard MDA methods use alternating optimization heuristics involving the computation of a succession of Tensor-matrix products. Such approaches are most of the time difficult to solve and not natural, highligthing the difficulty to formulate this problem in fully Tensor form. In this paper, we propose to solve multilinear discriminant analysis (MDA) by using the concept of transform domain (TD) recently proposed in [15]. We show here that moving MDA to this specific transform domain make its resolution easier and more natural. More precisely, each frontal face of the transformed Tensor is processed independently to build a separate optimization sub-problems easier to solve. Next, the obtained solutions are converted into projective Tensors by inverse transform. By considering a large number of experiments, we show the effectiveness of our approach with respect to existing MDA methods.

Recently, we have used the symmetric bracket of vector fields, and developed the notion of the symmetric derivation. Using this machinery, we have defined the concept of symmetric curvature. This concept is natural and is related to the notions divergence and Laplacian of vector fields. This concept is also related to the derivations on the algebra of symmetric forms which has been discussed by the authors. We introduce a new class of geometric vector fields and prove some basic facts about them. We call these vector fields affinewise. By contraction of the symmetric curvature, we define two new curvatures which have direct relations to the notions of divergence, Laplacian, and the Ricci Tensor.

Tensor completion has numerous applications in digital image processing such as image recovery and video overlay. In this paper, we consider two new approaches to Tensor completion. Efficient low-rank Tensor with Tensor train and Tensor ring for image recovery, some basic concepts about Tensor algebra and completion problems are presented, after that Tensor completion based on the Tensor train and Tensor ring are offered and implemented on some examples for image recovery with different observed ratios. The results of these implementations are compared and final results are proposed.

The ordinary Tensor product of modules is defined using bilinear maps (bimorphisms), that are linear in each component. keeping this in mind, Linton and Banaschewski with Nelson defined and studied the Tensor product in an equational category and in a general (concrete) category K, respectively, using bimorphisms, that is, defined via the Hom-functor on K. Also, the so-called sesquilinear, or one and a half linear maps and the corresponding Tensor products generalize these notions for modules and vector spaces. In this paper, taking a concrete category K and an arbitrary subfunctor H of the functor U'=Hom o (Uop´U) rather than just the Hom-functor, where U is the underlying set functor on K, we generalize sesquilinearity to bivariation and study the related notions such as functional internal lifts, universal bivariants, Tensor products, and their interdependence.

In this paper, we propose some new preconditioners  for solving multilinear system $\mathcal{A}\mathbf{x}^{m-1}=\mathbf{b}$. These preconditioners are based on Tensor splitting. We also present some theorems for analyzing and convergence of the preconditioned  Jacobi-, Gauss-Seidel-, and SOR-type iterative methods. Numerical examples are presented to verify the efficiency of the proposed preconditioned methods.

Recently, infinite and finite dimensional generalized Hilbert Tensors have been introduced. In this paper, the authors further introduce infinite and finite dimensional generalized Cesaro Tensors as a generalization of Cesaro matrices and discuss the properties of these structured Tensors. Next, some  upper bounds of $Z_{1}$-spectral radius of generalized Cesaro Tensors  and  generalized Hilbert Tensors are given,  which improves the existing ones. Finally, we obtain conditions under which a generalized Cesaro Tensor is column sufficient Tensor.