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BackgroundGene expression may be regulated by the DNA methylation of regulatory elements in cis, distal, and trans regions. One method to evaluate the relationship between DNA methylation and gene expression is the mapping of expression quantitative trait methylation (eQTM) loci (also called expression associated CpG loci, eCpG). However, no open-source tools are available to provide eQTM mapping. In addition, eQTM mapping can involve a large number of comparisons which may prevent the analyses due to limitations of computational resources. Here, we describe Torch-eCpG, an open-source tool to perform eQTM mapping that includes an optimized implementation that can use the graphical processing unit (GPU) to reduce runtime.ResultsWe demonstrate the analyses using the tool are reproducible, up to 18 × faster using the GPU, and scale linearly with increasing methylation loci.ConclusionsTorch-eCpG is a fast, reliable, and scalable tool to perform eQTM mapping. Source code for Torch-eCpG is available at https://github.com/kordk/torch-ecpg .

The extremal problem of two types of high order spectra for graphs are considered, which are called r -adjacency spectrum and t -clique spectrum, respectively. In this paper, we obtain the maximum r -adjacency spectral radius of a K r + 1 minor-free graph of order n in the case 1 ≤ r ≤ 3 , which implies the Hadwiger's conjecture is true for 1 ≤ r ≤ 3. Moreover, an upper bound of the 3-clique spectral radius of a B k -free and K 2 , l -free graph G of order n is given, where B k is the graph consisting of k triangles sharing an edge. As a corollary of this result, we obtain an upper bound of the number of the triangles for G which improves a result of Alon and Shikhelman (2016). [ABSTRACT FROM AUTHOR]

In this paper, we investigate bounds of solution set of the polynomial complementarity problem. When a polynomial complementarity problem has a solution, we propose a lower bound of solution norm by entries of coefficient tensors of the polynomial. We prove that the proposing lower bound is larger than some existing lower bounds appeared in tensor complementarity problems and polynomial complementarity problems. When the solution set of a polynomial complementarity problem is nonempty, and the coefficient tensor of the leading term of the polynomial is an R 0 -tensor, we propose a new upper bound of solution norm of the polynomial complementarity problem by a quantity defining by an optimization problem. Furthermore, we prove that when coefficient tensors of the polynomial are partially symmetric, the proposing lower bound formula with respect to tensor tuples reaches the maximum value, and the proposing upper bound formula with respect to tensor tuples reaches the minimum value. Finally, by using such partial symmetry, we obtain bounds of solution norm by coefficients of the polynomial. [ABSTRACT FROM AUTHOR]

In this paper, we extend Hardy's inequality to infinite tensors. To do so, we introduce Cesàro tensors ℭ , and consider them as tensor maps from sequence spaces into tensor spaces. In fact, we prove inequalities of the form ∥ ℭ ⁢ x k ∥ t , 1 ≤ U ⁢ ∥ x ∥ l p k ( k = 1 , 2 ), where x is a sequence, ℭ ⁢ x k is a tensor, and ∥ ⋅ ∥ t , 1 , ∥ ⋅ ∥ l p are the tensor and sequence norms, respectively. The constant U is independent of x, and we seek the smallest possible value of U. [ABSTRACT FROM AUTHOR]

Plant hyperspectral images (HSIs) contain valuable information for agricultural disaster prediction, biomass estimation, and other applications. However, they also include a lot of irrelevant background information, which wastes storage resources. In this paper, we propose a novel recursive sub-tensor hyperspectral compressive sensing method for plant leaves. This method uses recursive sub-tensor compressive sensing to compress and reconstruct each arbitrary-shape leaf region, discarding a large amount of background information to achieve the best possible reconstruction performance of the leaf region and significantly reduce storage space. The proposed method involves several key steps. Firstly, the optimal band is determined using the spatial spectral decorrelation criterion, and its corresponding mask image is used to extract the leaf regions from the background. Secondly, the recursive maximum inscribed rectangle algorithm is applied to obtain rectangular sub-tensors of leaves recursively. Each sub-tensor is then individually compressed and reconstructed. Finally, all sub-tensors can be reconstructed to form complete leaf HSIs without background information. Experimental results demonstrate that the proposed method achieves superior image reconstruction quality at extremely low sampling rates compared to other methods. The proposed method can improve average Peak Signal-to-Noise Ratio (PSNR) values by about 3.04% and 0.74% compared to Tensor Compressive Sensing (TCS) at the sampling rate of 2%. In the spectral domain, the proposed method can achieve significantly smaller Spectral Angle Mapper (SAM) values and relatively lower spectral indices errors for Double Difference, Triangular Vegetation Index, Leaf Chlorophyll Index, and Modified Normalized Difference 680 than those of TCS. Therefore, the proposed method achieves better compression performance for reconstructed plant leaf HSIs than the other methods. [ABSTRACT FROM AUTHOR]

The scale and complexity of big data are growing continuously, posing severe challenges to traditional data processing methods, especially in the field of clustering analysis. To address this issue, this paper introduces a new method named Big Data Tensor Multi-Cluster Distributed Incremental Update (BDTMCDIncreUpdate), which combines distributed computing, storage technology, and incremental update techniques to provide an efficient and effective means for clustering analysis. Firstly, the original dataset is divided into multiple sub-blocks, and distributed computing resources are utilized to process the sub-blocks in parallel, enhancing efficiency. Then, initial clustering is performed on each sub-block using tensor-based multi-clustering techniques to obtain preliminary results. When new data arrives, incremental update technology is employed to update the core tensor and factor matrix, ensuring that the clustering model can adapt to changes in data. Finally, by combining the updated core tensor and factor matrix with historical computational results, refined clustering results are obtained, achieving real-time adaptation to dynamic data. Through experimental simulation on the Aminer dataset, the BDTMCDIncreUpdate method has demonstrated outstanding performance in terms of accuracy (ACC) and normalized mutual information (NMI) metrics, achieving an accuracy rate of 90% and an NMI score of 0.85, which outperforms existing methods such as TClusInitUpdate and TKLClusUpdate in most scenarios. Therefore, the BDTMCDIncreUpdate method offers an innovative solution to the field of big data analysis, integrating distributed computing, incremental updates, and tensor-based multi-clustering techniques. It not only improves the efficiency and scalability in processing large-scale high-dimensional datasets but also has been validated for its effectiveness and accuracy through experiments. This method shows great potential in real-world applications where dynamic data growth is common, and it is of significant importance for advancing the development of data analysis technology. [ABSTRACT FROM AUTHOR]

There is growing interest to extend low-rank matrix decompositions to multi-way arrays, or tensors. One fundamental low-rank tensor decomposition is the canonical polyadic decomposition (CPD). The challenge of fitting a low-rank, nonnegative CPD model to Poisson-distributed count data is of particular interest. Several popular algorithms use local search methods to approximate the maximum likelihood estimator (MLE) of the Poisson CPD model. This work presents two new algorithms that extend state-of-the-art local methods for Poisson CPD. Hybrid GCP-CPAPR combines Generalized Canonical Decomposition (GCP) with stochastic optimization and CP Alternating Poisson Regression (CPAPR), a deterministic algorithm, to increase the probability of converging to the MLE over either method used alone. Restarted CPAPR with SVDrop uses a heuristic based on the singular values of the CPD model unfoldings to identify convergence toward optimizers that are not the MLE and restarts within the feasible domain of the optimization problem, thus reducing overall computational cost when using a multi-start strategy. We provide empirical evidence that indicates our approaches outperform existing methods with respect to converging to the Poisson CPD MLE. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel four-dimensional (4D) parameter estimation method to localize the mixed far-field (FF) and near-field (NF) targets using bistatic MIMO arrays and higher-order singular value decomposition (HOSVD). The estimated four parameters include the angle-of-departure (AOD), angle-of-arrival (AOA), range-of-departure (ROD), and range-of-arrival (ROA). In the method, we store array data in a tensor form to preserve the inherent multidimensional properties of the array data. First, the observation data are arranged into a third-order tensor and its covariance tensor is calculated. Then, the HOSVD of the covariance tensor is performed. From the left singular vector matrices of the corresponding module expansion of the covariance tensor, the subspaces with respect to transmit and receive arrays are obtained, respectively. The AOD and AOA of the mixed FF and NF targets are estimated with signal-subspace, and the ROD and ROA of the NF targets are achieved using noise-subspace. Finally, the estimated four parameters are matched via a pairing method. The Cramér–Rao lower bound (CRLB) of the mixed target parameters is also derived. The numerical simulations demonstrate the superiority of the tensor-based method. [ABSTRACT FROM AUTHOR]

The “field tensor” is the tensor product of the electric fields of the interacting waves during a sum‐ or difference‐frequency generation nonlinear optical interaction. It is therefore a tensor describing light interacting with matter, the latter being characterized by the “electric susceptibility tensor.” The contracted product of these two tensors of equal rank gives the light‐matter interaction energy, whether or not propagation occurs. This notion having been explicitly or implicitly present from the early pioneering studies in nonlinear optics, its practical use has led to original developments in many highly topical theoretical or experimental situations, at the microscopic as well macroscopic level throughout a variety of coherent or non‐coherent processes. The aim of this review article is to rigorously explain the field tensor formalism in the context of tensor algebra and nonlinear optics in terms of a general time‐space multi‐convolutional development, using spherical tensors, with components expressed in the frame of a common basis set of irreducible tensors, or Cartesian tensors. A wide variety of media are considered, including biological tissues and their imaging, artificially engineered by various combinations of optical and static electric fields, with the two extremes of all‐optical and purely electric poling, and also bulk single crystals. [ABSTRACT FROM AUTHOR]

The anisotropic characteristics of thermal deformation of ultrapure 2,4-dinitroanisole (2,4-DNAN) crystals were determined by the methods of powder thermorentgenography of the internal standard. The points of structural changes are registered in increments of 10 K, and in the melting region of 2 and 1 K. Calculations of powder X-ray diffraction data are performed by methods of full-profile analysis with a cycle of quantum modeling of the structure of molecules integrated into the algorithm. The Pauli, Le Bail (WPPD), Rietveld (WPPF) and WPPM methods were used as reference methods for full-profile analysis. The main crystallographic axes and characteristic surfaces of the thermal deformation tensor α and β-2,4-DNAN are determined. At atmospheric pressure, the main coefficients of linear (α) and volumetric (β) thermal deformation (expansion) were at 293 K for α-2,4-DNAN with α1(293) = 11,516 x 10-5 K-1, α2(293) = - 0,120 x 10-5 K-1, α3(293) = 5,098 x 10-5 K-1, β(293) = 16,333 x 10-5 K-1; at 293 K for β-2,4-DNAN with α1(293) = 13,217 x 10-5 K-1, α2(293) = 0,494 x 10-5 K-1, α3(293) = -8,6504 x 10-5 K-1, β(293) = 6,8191 x 10-5 K-1; at 260 K for β'-2,4-DNAN with α1(260) = 25,214 x 10-5 K-1, α2(260) = -5,823 x 10-5 K-1, α3(260) = 7,741 x 10-5 K-1, β(260) = 27,112 x 10-5 K-1. [ABSTRACT FROM AUTHOR]

To address the timely acquisition of channel state information in reconfigurable intelligent surface (RIS)-assisted millimetre wave (mmWave) multiple-input multiple-output (MIMO) systems, a channel estimation scheme based on tensor decomposition was proposed. Firstly, a channel training mechanism with low pilot overhead was designed using a few passive reflection units and constructing a phase shift matrix. Then, a non-iterative channel estimation algorithm was derived using tensor canonical polyadic decomposition with Vandermonde structure constraints. Theoretical analysis indicated that the minimum pilot overhead of the proposed scheme only depended on the product of the subchannel path numbers of the reflection links and exhibited low computational complexity. Simulation results further verify the superiority of the proposed scheme compared to other methods. [ABSTRACT FROM AUTHOR]

Purpose:: Considering successful clinical outcomes, accurate osteotomy/implantation and soft tissue balancing are essential in total knee arthroplasty (TKA). However, intra-operative assessment of soft tissue balance remains difficult, and management is left much to the surgeon's subjective feel and experience. The aim of this paper was to review various soft tissue balance assessments and their relationship with pre- and intra-operative factors and clinical outcomes. Methods:: Literature regarding the history of soft tissue balance measurement, various types of measurement tools, theory of recent measurement, influence of various factors on soft tissue balance, and influence of soft tissue balance on clinical outcomes in TKA was reviewed using the PubMed database. Results:: Soft tissue balance measurement has switched from the unphysiological condition, i.e., with assessment between bone cut surfaces and patellar eversion, to the physiological condition, i.e. with femoral component placement and patellofemoral joint reduction. Type of prosthesis, implant design, surgical technique, and pre-operative factors affect intra-operative soft tissue balance. Intra-operative soft tissue balance also affects post-operative range of motion and patient-reported outcome measures. Conclusions:: Intra-operative quantitative soft tissue balance measurement and management with physiological knee condition, which is closely influenced by various pre-operative and intra-operative factors, is important for the achievement of high knee function and patient satisfaction.

Optimal network analysis requires advanced techniques to handle the inherent complexity and uncertainty of real-world systems. We have used vertex order coloring on neutrosophic graphs to find the most effective approach to improve network reliability and performance. Neutrosophic graphs(NG) offer a comprehensive framework for modelling real-world networks with inherent uncertainties by incorporating degrees of truth, falsity, and indeterminacy. In this paper, we have investigated various graph product operations as a means of optimizing network structures. We further investigated the applications of vertex order coloring to identify α, β and γ strong vertices within various graph operations of NG. We examined several NG products with the goal of determining the most optimal network based on particular important metrics including the total number of alpha-strong vertices, the weight of alpha-strong vertices, the chromatic number, and the weight of the graph's minimum spanning tree. The objective of our research is to identify the best solutions that strike a balance between robustness and association by rigorously studying and comparing various product operations. Our research advances the subject of network theory and provides useful information for a variety of applications, including social networks, transportation, and telecommunications. [ABSTRACT FROM AUTHOR]

Let r and s be positive integers such that r ⩾ 3. Let U 1 , ... , U r be vector spaces over a field F and V 1 , ... , V s be vector spaces over a field K such that dim ⁡ U i , dim ⁡ V j ⩾ 2 for all i , j. In this paper, we characterize maps ψ : ⨂ i = 1 r U i → ⨂ i = 1 s V i that preserve adjacency in both directions, which extends Hua's fundamental theorem of geometry of rectangular matrices. We also characterize related results concerning locally full maps preserving adjacency in both directions between tensor spaces, maps preserving adjacency in both directions between tensor spaces over a field all whose nonzero endomorphisms are automorphisms, and injective continuous adjacency preserving maps on finite dimensional tensor spaces over the real field. [ABSTRACT FROM AUTHOR]

Eddy viscosity is employed throughout the majority of numerical fluid dynamical models, and has been the subject of a vigorous body of research spanning a variety of disciplines. It has long been recognized that the proper description of eddy viscosity uses tensor mathematics, but in practice it is almost always employed as a scalar due to uncertainty about how to constrain the extra degrees of freedom and physical properties of its tensorial form. This manuscript borrows techniques from outside the realm of geophysical fluid dynamics to consider the eddy viscosity tensor using its eigenvalues and eigenvectors, establishing a new framework by which tensorial eddy viscosity can be tested. This is made possible by a careful analysis of an operation called tensor unrolling, which casts the eigenvalue problem for a fourth‐order tensor into a more familiar matrix‐vector form, whereby it becomes far easier to understand and manipulate. New constraints are established for the eddy viscosity coefficients that are guaranteed to result in energy dissipation, backscatter, or a combination of both. Finally, a testing protocol is developed by which tensorial eddy viscosity can be systematically evaluated across a wide range of fluid regimes. Plain Language Summary: Numerical fluid flow solvers need to dissipate energy in order to remain numerically stable, and this is most often achieved by adding a mechanism to the governing equations called eddy viscosity. Generally the implementation of eddy viscosity boils down to specifying a scalar coefficient that governs the rate of energy dissipation. However, the true mathematical form of eddy viscosity is that of a higher‐order geometric object called a tensor, and the potential advantages of using this form remain unexplored. This paper uses a generalized version of familiar linear algebra operations (eigenvalues, trace, and determinant) to establish new constraints on the eddy viscosity coefficients that promise to open up this parameterization to renewed scrutiny. Key Points: Eddy viscosity is usually employed as a scalar coefficient, but its true form is that of a tensorEigenanalysis can reveal new constraints on the coefficients of the eddy viscosity tensorTensor unrolling can help expose the power of the eigenanalysis, but only if done in a particular way [ABSTRACT FROM AUTHOR]

This research work extends the least squares criterion. The regression models which have been treated so far in the literature do not study multilinear relationships between variables. Such relationships are of a nonlinear nature. They take place whenever two or more than two univariate variables are the components of a multiple variable of order 2 or an order greater than 2. A multiple variable of order 2 is not a bivariate variable, and a multiple variable of an order greater than 2 is not a multivariate variable. A multiple variable allows for the construction of a tensor. The α -norm of this tensor gives rise to an aggregate measure of a multilinear nature. In particular, given a multiple variable of order 2, four regression lines can be estimated in the same subset of a two-dimensional linear space over R. How these four regression lines give rise to an aggregate measure of a multilinear nature is shown by this paper. In this research work, such a measure is an estimate concerning the expected return on a portfolio of financial assets. The metric notion of α -product is used to summarize the sampling units which are observed. [ABSTRACT FROM AUTHOR]

The main aim of this article is to introduce the weighted numerical range and the weighted numerical radius for an even-order square tensor via the Einstein product and establish their various properties. Also, the proof of convexity of the numerical range of a tensor is revisited. The notions of weighted unitary tensor, weighted positive definite tensor, and weighted positive semi-definite tensor are then discussed. The spectral decomposition for normal tensors is also provided. This is then used to present the equality between the weighted numerical radius and the spectral radius of a weighted normal tensor. As applications of the above fact, a few equalities of weighted numerical radius and weighted tensor norm are obtained. [ABSTRACT FROM AUTHOR]

It is well known that a complex circulant matrix can be diagonalized by a discrete Fourier matrix with imaginary unit i. The main aim of this paper is to demonstrate that a quaternion circulant matrix cannot be diagonalized by a discrete quaternion Fourier matrix with three imaginary units i, j, and k. Instead, a quaternion circulant matrix can be block-diagonalized into 1-by-1 block and 2-by-2 block matrices by permuted discrete quaternion Fourier transform matrix. With such a block-diagonalized form, the inverse of a quaternion circulant matrix can be determined efficiently similarly to the inverse of a complex circulant matrix. We make use of this block-diagonalized form to study quaternion tensor singular value decomposition of quaternion tensors where the entries are quaternion numbers. The applications, including computing the inverse of a quaternion circulant matrix and solving quaternion Toeplitz systems arising from linear prediction of quaternion signals, are employed to validate the efficiency of our proposed block-diagonalized results. [ABSTRACT FROM AUTHOR]

Hyperspectral images (HSIs) possess an inherent three-order structure, prompting increased interest in extracting 3D features. Tensor analysis and low-rank representations, notably truncated higher-order SVD (T-HOSVD), have gained prominence for this purpose. However, determining the optimal order and addressing sensitivity to changes in data distribution remain challenging. To tackle these issues, this paper introduces an unsupervised Superpixelwise Multiscale Adaptive T-HOSVD (SmaT-HOSVD) method. Leveraging superpixel segmentation, the algorithm identifies homogeneous regions, facilitating the extraction of local features to enhance spatial contextual information within the image. Subsequently, T-HOSVD is adaptively applied to the obtained superpixel blocks for feature extraction and fusion across different scales. SmaT-HOSVD harnesses superpixel blocks and low-rank representations to extract 3D features, effectively capturing both spectral and spatial information of HSIs. By integrating optimal-rank estimation and multiscale fusion strategies, it acquires more comprehensive low-rank information and mitigates sensitivity to data variations. Notably, when trained on subsets comprising 2%, 1%, and 1% of the Indian Pines, University of Pavia, and Salinas datasets, respectively, SmaT-HOSVD achieves impressive overall accuracies of 93.31%, 97.21%, and 99.25%, while maintaining excellent efficiency. Future research will explore SmaT-HOSVD's applicability in deep-sea HSI classification and pursue additional avenues for advancing the field. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel rank-adaptive higher-order orthogonal iteration (RA-HOOI) algorithm to solve the fixed-accuracy low multilinear-rank approximation of tensors. On the one hand, RA-HOOI relies on a greedy strategy to expand the subspace, which avoids computing the full SVD of the matricization of the input tensor. On the other hand, the new rank-adaptive strategy introduced in the RA-HOOI algorithm enables the obtained truncation to be more accurate. A series of numerical experiments related to synthetic and real-world tensors are carried out to show that the proposed RA-HOOI algorithm is comparable to state-of-the-art methods in terms of both accuracy and efficiency and performs better in certain situations. [ABSTRACT FROM AUTHOR]

This paper establishes the solvability conditions and an expression of the exact solution to a system of three Sylvester-like quaternion tensor equations in four variables. Based on a comprehensive analysis of the general solution and the solvability conditions associated with the system, necessary and sufficient conditions are deduced to a system of Sylvester-like tensor equations, including the unknowns as η-Hermitian quaternion tensors. Ultimately, we design an algorithm to compute the general solution, even a numerical example to illustrate the essential findings of this paper. [ABSTRACT FROM AUTHOR]

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In this paper, we present two multidimensional power flow formulations based on a fixed-point iteration (FPI) algorithm to efficiently solve hundreds of thousands of Power flows (PFs) in distribution systems. The presented algorithms are the base for a new TensorPowerFlow (TPF) tool and shine for their simplicity, benefiting from multicore Central processing unit (CPU) and Graphics processing unit (GPU) parallelization. We also focus on the mathematical convergence properties of the algorithm, showing that its unique solution is at the practical operational point. The proof is validated using numerical simulations showing the robustness of the FPI algorithm compared to the classical Newton–Raphson (NR) approach. In the case study, a benchmark with different PF solution methods is performed, showing that for applications requiring a yearly simulation at 1-minute resolution, the computation time is decreased by a factor of 164, compared to the NR in its sparse formulation. Finally, a set of applications is described, highlighting the potential of the proposed formulations over a wide range of analyses in distribution systems.

Plant hyperspectral images (HSIs) contain valuable information for agricultural disaster prediction, biomass estimation, and other applications. However, they also include a lot of irrelevant background information, which wastes storage resources. In this paper, we propose a novel recursive sub-tensor hyperspectral compressive sensing method for plant leaves. This method uses recursive sub-tensor compressive sensing to compress and reconstruct each arbitrary-shape leaf region, discarding a large amount of background information to achieve the best possible reconstruction performance of the leaf region and significantly reduce storage space. The proposed method involves several key steps. Firstly, the optimal band is determined using the spatial spectral decorrelation criterion, and its corresponding mask image is used to extract the leaf regions from the background. Secondly, the recursive maximum inscribed rectangle algorithm is applied to obtain rectangular sub-tensors of leaves recursively. Each sub-tensor is then individually compressed and reconstructed. Finally, all sub-tensors can be reconstructed to form complete leaf HSIs without background information. Experimental results demonstrate that the proposed method achieves superior image reconstruction quality at extremely low sampling rates compared to other methods. The proposed method can improve average Peak Signal-to-Noise Ratio (PSNR) values by about 3.04% and 0.74% compared to Tensor Compressive Sensing (TCS) at the sampling rate of 2%. In the spectral domain, the proposed method can achieve significantly smaller Spectral Angle Mapper (SAM) values and relatively lower spectral indices errors for Double Difference, Triangular Vegetation Index, Leaf Chlorophyll Index, and Modified Normalized Difference 680 than those of TCS. Therefore, the proposed method achieves better compression performance for reconstructed plant leaf HSIs than the other methods.

In this paper, we consider a general tensor complementarity problem with interval parameters, and study the conditions under which, the existence and uniqueness of the solution of the problem are guaranteed. Furthermore, we proved that the solution set of the interval tensor complementarity problem is not necessarily convex. 1. IntroductionInterval analysis is a branch of numerical analysis that was born in the 1960's. It consists of computing with intervals of reals instead of reals, providing a framework for handling uncertainties and verified computations. The result of an interval computation is an interval, a pair of numbers, an upper and a lower bound, and this pair of numbers guarantees to enclose the exact answer. Maybe we still don’t know the truth, but at least we know how much we don’t know! [4]. How is it possible for interval analysis to guarantee that a computational result is true? The answer is very simple. Using the interval analysis we estimate at each calculation step all kinds of errors: inputs errors, rounding errors and truncation errors. One of the most famous references on IA is probably Moore’s Interval Analysis book. Throughout the paper, vectors are written as $\left\{x, y, \ldots \right\}$, matrices are shown by $ \left\{A, B,\ldots \right\}$ and tensors are written as $ \left\{ \mathcal{A}, \mathcal{B},\ldots \right\}.$ Let $[n]$, $\mathbb{R} (\mathbb{C}) $, and $\mathbb{R}^n (\mathbb{C}^n)$ denote the set $\{ 1, 2,\ldots,n\}$, the set of all real (complex) numbers, and the set of all $n$-dimensional real (complex) vectors; respectively. $x \geq 0 \ \ \ (x > 0)$ means $x_i \geq 0 \ \ \ (x_i > 0)$ for all $i \in \left[ n \right]$. Let $\mathbb{R} _{+}^n = \lbrace x \in \mathbb{R}^n \mid x \geq 0\rbrace$ be the positive cone in $\mathbb{R}^n.$ An order $m$ dimension $n$ real tensor $\mathcal{A} = (a_{i_1 i_2\cdots i_m}),$ denoted by $ \mathcal{A} \in \mathbb{R}^{n_1 \times \cdots \times n_m } ,$ consists of $n^m$ entries: \[a_{i_1 i_2 \cdots i_m} \in \mathbb{R}, \quad \; \forall \; i_j = 1,\cdots,n ,\quad j = 1,\cdots,m.\] If $n_1=\cdots=n_m = n$, then it is said $\mathcal{A}$ is an $m$-order $n$-dimensional cubical tensor or for simplicity just $m$-order $n$-dimensional tensor. A vector is a tensor of order $1$ and a matrix is a tensor of order $2$. A tensor $\mathcal{A} = (a_{i_1 i_2 \cdots i_m}) \in \mathbb{R}^{n_1 \times \cdots \times n_m } $ is called nonnegative (positive) if \[a_{i_1 i_2 \cdots i_m} \ge 0 \; (a_{i_1 i_2 \cdots i_m}>0 ), \quad \; \forall \; i_j = 1,\cdots,n ,\quad j = 1,\cdots,m.\] A tensor $ \mathcal{A}$ is said to be symmetric if its entries $ a_{i_1 i_2 \cdots i_m}$ are invariant under any permutation of $ m $ indices $ ( a_{i_1 i_2 \cdots i_m}). $ All the tensors discussed in this paper are real.\\For any two tensors, $\mathcal{A} = (a_{i_1 \cdots i_m } ),$ and $ \mathcal{B} = (b_{i_1 \cdots i_m } ) \in \mathbb{R}^{n_1 \times \cdots \times n_m } $ of identical orders and dimensions, their inner product is defined as $$\left\langle {\mathcal{A},\mathcal{B}} \right\rangle = \sum\limits_{i_1 \cdots i_m } {a_{i_1 \cdots i_m } b_{i_1 \cdots i_m}}. $$ Definition 1.1. If $\mathcal{A} \in \mathbb{R}^{n_1 \times \cdots \times n_m } $ is an $m$-order tensor and $ B \in \mathbb{R}^{J \times n_k }$ is a matrix, then $\mathcal{A} \times_k B $ denotes the mode-$k$ product of $\mathcal{A}$ with $B$, which is of size $ n_1 \times \cdots \times n_{k-1} \times J \times n_{k+1} \times \cdots \times n_m $, and each element of it is defined as follows \[(\mathcal{A} \times_k B)_{i_1,\ldots,i_{k-1}, j, i_{k+1},\ldots,i_m} = \sum\limits_{i_k = 1}^{n_k } a_{i_1 \cdots i_m } b_{j,i_k}.\]If we do the mode-$k$ product of $ \mathcal{A} $ and $B $ for all possible $k \in [m]$ as \[ \mathcal{A} \times_1 B \times_2 \cdots \times_m B,\] and $B $ is reduced to some row vector, say $x^T=\left( x_1,\ldots,x_n \right),$ the following notations are frequently used in this paper:\begin{align*}&\mathcal{A} x^m \equiv \mathcal{A} \times_1 x^T \times_2 \cdots \times_m x^T = \sum\limits_{i_1 \cdots i_m =1}^n {a_{i_1 \cdots i_m} x_{i_1} \cdots x_{i_m}} \in \mathbb{R}, \\&\mathcal{A} x^{m-1 } \equiv \mathcal{A} \times_2 x^T \times_3 \cdots \times_m x^T = \sum\limits_{i_2 \cdots i_m =1}^n {a_{i, i_2 \cdots i_m} x_{i_2} \cdots x_{i_m}} \in \mathbb{R}^n.\end{align*}We call a number $\lambda \in \mathbb{C}$ an eigenvalue of $\mathcal{A}$ if it and a nonzero vector $x \in \mathbb{C}^n$ are solutions of the following homogeneous polynomial equations:(1.1)\begin{equation}\label{e3}\left( \mathcal{A} x^{m-1} \right)_i = \lambda x_i^{m - 1}, \ \ \ \forall i=1,\ldots,n,\end{equation}and call the solution $x$ an eigenvector of $\mathcal{A}$ associated with the eigenvalue $\lambda.$ If we denote $x^{[m-1]}$ as a vector in $\mathbb{C}^n$ such that its $i$th component is $x_i^{m - 1},$ then (1.1) can be simply expressed as $$\mathcal{A} x^{m-1} = \lambda x^{[m-1]}.$$ The set of all the eigenvalues of $\mathcal{A}$ is called the spectrum of $\mathcal{A}.$ The largest modulus of the elements in the spectrum of $\mathcal{A}$ is called the spectral radius of $\mathcal{A},$ denoted as $\rho(\mathcal{A}).$ Definition 1.2. Let $\mathcal{A}$ be an $m$-order and $n$-dimensional cubical tensor, then$(1)$ $\mathcal{A}$ is called an $\mathcal{Z}$-tensor if all of its non-diagonal elements are non-positive. This definition is equivalent to having $\mathcal{A}=s\mathcal{I}-\mathcal{B}$, where $s > 0$, $\mathcal{B}$ is a non-negative tensor and $\mathcal{I}=(I_{i_1 \cdots i_m })$, is the identity tensor with entries$$I_{i_1 \cdots i_m}=\left\{ \begin{gathered}1, \ \ \ i_1=\cdots= i_m,\hfill \\0, \ \ \ otherwise. \hfill \\\end{gathered} \right.$$$(2)$ $\mathcal{A}$ is called an $\mathcal{M}$-tensor if $\mathcal{A}$ is an $\mathcal{Z}$-tensor and $\mathcal{A} = s\mathcal{I} - \mathcal{B}$, $s \geq \rho(\mathcal{B})$. If $s > \rho(B)$, then $\mathcal{A}$ is called a strong(nonsingular) $\mathcal{M}$-tensor. Definition 1.3. An $m$-order $n$-dimensional tensor $\mathcal{A}$ is said to be P-tensor, if for each nonzero $x \in \mathbb{R}^n $, there exists some index $i$such that(1.2)\begin{equation}\label{e4}x_i \left( {\mathcal{A}x^{m - 1}} \right)_i >0.\end{equation}The tensor $\mathcal{M}(\mathcal{A} )=(m_{i_1 \cdots i_m})$ is called the comparison tensor of $\mathcal{A}$ if$$ m_{i_1 \cdots i_m}= \begin{cases}-\left|a_{i_1 \cdots i_m}\right|, & \text { if } (i_2,\cdots,i_m) \neq (i_1,\cdots,i_1) \\ \left|a_{i_1 \cdots i_m}\right|, & \text { if } (i_2,\cdots,i_m)=(i_1,\cdots,i_1)\end{cases} $$ Definition 1.4. A tensor $\mathcal{A}$ is called an $\mathcal{H}$-tensor, if its comparison tensor is an $\mathcal{M}$-tensor, and it is called a nonsingular $\mathcal{H}$ -tensor, if its comparison tensor is nonsingular. The tensor complementarity problem denoted by the $\operatorname{TCP}(\mathbf{q}, \mathcal{A})$, is to find a vector $\mathbf{x}$ such that:$$ \mathbf{x} \geq 0, ~\mathcal{A} \mathbf{x}^{m-1}+\mathbf{q} \geq 0,~\left\langle\mathbf{x}, \mathcal{A} \mathbf{x}^{m-1}+\mathbf{q}\right\rangle=0, $$ where $\left\langle , \right\rangle $ denotes the inner product. Theorem 1.5. [6] Tensor $\mathcal{A}$ is $P$-tensor if and only if TCP(q, $\mathcal{A}$) has a unique solution for every $q > 0.$ 2. Main ResultsInterval linear algebra is a mathematical field developed from classical linear algebra. The only difference is that we do not work with real numbers but we deal with the real closed intervals $x^I : = \left[ {\underline{x}, \overline{x}} \right]$, where $\underline{x} \leq \overline{x}.$ An interval tensor is a tensor which every of its elements is interval. An $m$-order $n$-dimensional cubical interval tensor is denoted by $\mathcal{A}^I : = \left[{\underline{\mathcal{A}}, \overline{\mathcal{A}} } \right]$, where${\underline{ \mathcal{A}}}$ and ${\overline {\mathcal{A}}}$ are real tensors, and \[\mathcal{A}^I (i_1,\cdots,i_m ) = \left[ {\underline {\mathcal{A}} (i_1,\cdots,i_m), \overline {\mathcal{A}} (i_1,\cdots,i_m )} \right].\] The set of all interval tensors of size $ n_1 \times \cdots \times n_m$, is denoted by $\mathbb{I}\mathbb{R}^{n_1 \times \cdots \times n_m}. $ The parametric form of interval, interval vector and interval tensor can be expressed as follows, respectively. $${a^I=\left\{a(t) \in \mathbb{R}: a(t)=\underline{a}+t\left(\overline{a}-\underline{a}\right), \underline{a} \in \mathbb{R}, \overline{a} \in \mathbb{R}, t \in[0,1]\right\},}$$ $${x^I=\left\{x(t) \in \mathbb{R}^{n}: x(t)=\underline{x}+t\left(\overline{x}-\underline{x}\right), \underline{x} \in \mathbb{R}^{n}, \overline{x} \in \mathbb{R}^{n}, t \in[0,1]\right\},} $$ $${\mathcal{A}^I=\left\{\mathcal{A}(t) \in \mathbb{R}^{n \times n \cdots \times n}: \mathcal{A}(t)=\underline{\mathcal{A}}+t\left(\overline{\mathcal{A}}-\underline{\mathcal{A}}\right), \underline{\mathcal{A}} \in \mathbb{R}^{n \times n \cdots \times n}, \overline{\mathcal{A}} \in \mathbb{R}^{n \times n \cdots \times n}, t \in[0,1]\right\}.}$$ Definition 2.1. For $\mathcal{A}^I=[a_{i_1 \cdots i_m}] \in \mathbb{I R}^{n \times n \cdots \times n}$, we define the comparison tensor of $\mathcal{A}^I$ and it is represented as $\mathcal{M}(\mathcal{A}^I )=(m_{i_1 \cdots i_m}) \in \mathbb{R}^{n \times n \cdots \times n}$ by setting$$ m_{i_1 \cdots i_m}= \begin{cases}-\left|[a_{i_1 \cdots i_m}]\right|, & \text { if } (i_2,\cdots,i_m) \neq (i_1,\cdots,i_1) \\ \left|[a_{i_1 \cdots i_m}]\right|, & \text { if } (i_2,\cdots,i_m)=(i_1,\cdots,i_1)\end{cases}$$ Definition 2.2. An interval tensor $\mathcal{A}^I \in \mathbb{I R}^{n \times n \cdots \times n}$ is called an interval $p$-tensor, $\mathcal{Z}$-tensor, $\mathcal{M}$-tensor and $\mathcal{H}$-tensor, if all $\mathcal{A} \in\mathcal{A}^I$ are $p$-tensor, $\mathcal{Z}$-tensor, $\mathcal{M}$-tensor and $\mathcal{H}$-tensor, respectively. Let $\mathcal{A}^I $ be an $m$-order and $n$-dimensional interval tensor and $q^I \in \mathbb{I R}^{n}$ be an $n$ dimensional interval vector. Then we consider the family of $T CP(\mathcal{A}, q)$'s$$q+\mathcal{A} x^{m - 1} \geq 0, \quad x \geq 0, x^ T ( q+\mathcal{A} x^{m - 1} ) = 0, ~\text {where}~ \mathcal{A} \in\mathcal{A}^I, q^I$$This is equivalent to the following family of $T C P(\mathcal{A}, q)$'s$$w-\mathcal{A} x^{m - 1}=q, \quad x \geq 0, w \geq 0, x^ T w=0, \text { where } \mathcal{A} \in\mathcal{A}^I, q^I.$$The family of $T C P(\mathcal{A}, q)$'s is represented as interval tensor complementarity problem and it is denoted by $\operatorname{ITCP}(\mathcal{A}^I, q^I) \cdot \sum_{x}(\mathcal{A}^I,q^I)$ is denoted as solutions set of $\operatorname{ITCP}(\mathcal{A}^I, q^I)$ and it is defined as $$\left\{x \in \mathbb{R}^{n}: q+ \mathcal{A} x^{m - 1} \geq 0, x \geq 0, x^ T (q+ \mathcal{A} x^{m-1}) =0, \mathcal{A} \in\mathcal{A}^I, q^I\right\}$$ The parametric form of $\operatorname{ITCP}(\mathcal{A}^I,q^I)$ is represented as $T C P(\mathcal{A}(t), q(t)), t \in[0,1]$ and its solution set is defined as for some fixed $t,$$$\sum_{x}(\mathcal{A}(t), q(t))=\left\{x \in \mathbb{R}^{n}: q(t)+\mathcal{A}(t) x^{m - 1} \geq 0, x \geq 0,x^{T} (q(t)+\mathcal{A}(t) x^{m - 1}) =0\right\}$$ Lemma 2.3. For any $\operatorname{ITCP}(\mathcal{A}^I,q^I), \sum_{x}\left(\underline{\mathcal{A}}, \underline{q}\right)$ and $\sum_{x}\left(\overline{\mathcal{A}}, \overline{q}\right)$ are subsets of $\sum_{x}(\mathcal{A}^I,q^I)$. Theorem 2.4. Suppose that $T C P\left(\underline{\mathcal{A}}, \underline{q}\right)$ and $T C P\left(\overline{\mathcal{A}}, \overline{q}\right)$ have unique solutions, then $\sum_{x}(\mathcal{A}(t), q(t))$ is a singleton set, for every $t \in[0,1]$. Proof. Suppose that $T C P\left(\underline{\mathcal{A}}, \underline{q}\right)$ and $T C P\left(\overline{\mathcal{A}}, \overline{q}\right)$ have unique solutions, then from Theorem \ref{pp} we get that $\underline{\mathcal{A}}$ and $\overline{\mathcal{A}}$ are $P$-tensors. Since the positive convex combination of $P$-tensors is also a $P$-tensor, then $\mathcal{A}(t)=t \underline{\mathcal{A}}+(1-t) \overline{\mathcal{A}}$, $t \in[0,1]$, is a $P$-tensor. Hence $\operatorname{TCP}(\mathcal{A}(t), q(t))$ has unique solution for each $t$. Therefore, $\sum_{x}(\mathcal{A}(t), q(t))$ is a singleton set, for every $t \in[0,1]$.Theorem 2.5. The set $\sum_{(w, x)}(\mathcal{A}^I,q^I)$ is not convex.Proof. Let $\left(w^{1}, x^{1}\right)^{T},\left(w^{2}, x^{2}\right)^{T} \in \sum_{(w, x)}(\mathcal{A}^I,q^I)$. Then,(2.1)$$\begin{gathered}& w_{i}^{j}-\sum_{i_2 \cdots i_m=1}^{n} \overline{a}_{i i_2 \cdots i_m} x_{i_2}^{j} \cdots x_{i_m}^{j} \leq \overline{q}_{i}, w_{i}^{j}-\sum_{i_2 \cdots i_m=1}^{n} \underline{a}_{i i_2 \cdots i_m} x_{i_2}^{j} \cdots x_{i_m}^{j} \geq \underline{q}_{i},\\ & x_{i_2}^{j} \cdots x_{i_m}^{j} \geq 0, w_{i}^{j} \geq 0, w_{i}^{j} x_{i}^{j}=0, i_1, \cdots, i_m=1, 2,\ldots,n, ~ j=1,2.\end{gathered}$$Let $0 \leq \lambda \leq 1$. Then multiplying (2.1) by $\lambda$ for $j=1$, and multiplying (2.1) by $(1-\lambda)$ for $j=2$, and adding up these inequalities, we get$$\begin{gathered}\left(\lambda w_{i}^{1}+(1-\lambda) w_{i}^{2}\right)-\sum_{i_2 \cdots i_m=1}^{n} \overline{a}_{i i_2 \cdots i_m}\left(\lambda x_{i_2}^{1} \cdots x_{i_m}^{1}+(1-\lambda) x_{i_2}^{2} \cdots x_{i_m}^{2}\right) \leq \overline{q}_{i}, i=1,2, \ldots, n, \\\left(\lambda w_{i}^{1}+(1-\lambda) w_{i}^{2}\right)-\sum_{i_2 \cdots i_m=1}^{n} \underline{a}_{i i_2 \cdots i_m}\left(\lambda x_{i_2}^{1} \cdots x_{i_m}^{1}+(1-\lambda) x_{i_2}^{2} \cdots x_{i_m}^{2}\right) \geq \underline{q}_{i}, i=1,2, \ldots, n, \\\left(\lambda x_{i_2}^{1} \cdots x_{i_m}^{1}+(1-\lambda) x_{i_2}^{2} \cdots x_{i_m}^{2}\right) \geq 0,\left(\lambda w_{i}^{1}+(1-\lambda) w_{i}^{2}\right) \geq 0, i_1,\cdots,i_m=1, 2,\ldots,n.\end{gathered}$$On the other hand, $\left(\lambda w_{i}^{1}+(1-\lambda) w_{i}^{2}\right)\left(\lambda x_{i_2}^{1} \cdots x_{i_m}^{1}+(1-\lambda) x_{i_2}^{2} \cdots x_{i_m}^{2}\right)=0$ only for $t=0$ and $t=1$. Since convex combination of complementarity variables are not complementarity variables. This gives that $\left(\lambda w^{1}+(1-\lambda) w^{2}, \lambda x^{1}+(1-\lambda) x^{2}\right)^{T} \notin \sum_{(w,x)}(\mathcal{A}^I,q^I)$, and so $\sum_{(w, x)}(\mathcal{A}^I,q^I)$ is not a convex set. Theorem 2.6. Let $\mathcal{A}^I $ be an $m$-order and $n$-dimensional interval $H$-tensor. Then, $\operatorname{ITCP}(\mathcal{A}^I,q^I)$ has a solution for every $q^I \in \mathbb{I R}^{n}$, that is, $\sum_{x}(\mathcal{A}^I,q^I)$ is a non-empty set. Proof. Let $\mathcal{A}^I $ be an $m$-order and $n$-dimensional interval $H$-tensor. Then each $\mathcal{A} \in\mathcal{A}^I$ is an $H$-tensor, and so $T C P(\mathcal{A}, q)$ has a solution. Also, $\sum_{x}(\mathcal{A}, q) \subseteq \sum_{x}(\mathcal{A}^I,q^I)$, which yields that $\sum_{x}(\mathcal{A}^I,q^I)$ is non-empty. 3. ConclusionA methodology is developed to discuss the existence of the solution of tensor complementarity problem, where the parameters are closed intervals. We also proved that the solution set of the interval tensor complementarity problem is not necessarily convex.

Background Back-projection has been used to correct the variance and polarity indeterminacies for the independent component analysis. The variance and polarity of the components are essential features of neuroscience studies.Objective This work extends the back-projection theory to canonical polyadic decomposition (CPD) for high-order tensors, aiming to correct the variance and polarity indeterminacies of the components extracted by CPD.Methods The tensor is reshaped into a matrix and decomposed using a suitable blind source separation algorithm. Subsequently, the coefficients are projected using back-projection theory, and other factor matrices are computed through a series of singular value decompositions of the back-projection matrix.Results By applying this method, the energy and polarity of each component are determined, effectively correcting the variance and polarity indeterminacies in CPD. The proposed method was validated using simulated tensor data and resting-state fMRI data.Conclusion Our proposed back-projection method for high-order tensors effectively corrects variance and polarity indeterminacies in CPD, offering a precise solution for calculating the energy and polarity required to extract meaningful features from neuroimaging data.

TensorView for MATLAB is a GUI-based visualization tool for depicting second-rank Cartesian tensors as surfaces on three-dimensional molecular models. Both ellipsoid and ovaloid tensor display formats are supported, and the software allows for easy conversion of Euler angles from common rotation schemes (active, passive, ZXZ, and ZYZ conventions) with visual feedback. In addition, the software displays all four orientation-equivalent Euler angle solutions for the placement of a single tensor in the molecular frame and can report relative orientations of two tensors with all 16 orientation-equivalent Euler angle sets that relate them. The salient relations are derived and illustrated through several examples. TensorView for MATLAB expands and complements the earlier implementation of TensorView within the Mathematica programming environment and can be run without a MATLAB license. TensorView for MATLAB is available through github at https://github.com/LeoSvenningsson/TensorViewforMatlab, and can also be accessed directly via the NMRbox resource.

The concept of $ C_{\pi }^R $ C π R -matrix is extended to $ C_{\pi }^R $ C π R -tensor, which also generalizes the concept of C-tensor. A necessary and sufficient condition for a tensor to be a $ C_{\pi }^R $ C π R -tensor is provided. We analyse decompositions of $ C_{\pi }^R $ C π R -tensors and prove that $ C_{\pi }^R $ C π R -tensors are nonsingular. Positive linear combinations and Hadamard product of two $ C_{\pi }^R $ C π R -tensors are also discussed. Finally, some properties of $ B_{\pi }^R $ B π R -tensor are given to localize real eigenvalues of a tensor. [ABSTRACT FROM AUTHOR]

In advanced manufacturing processes, high-dimensional (HD) streaming data (e.g., sequential images or videos) are commonly used to provide online measurements of product quality. Although there exist numerous research studies for monitoring and anomaly detection using HD streaming data, little research is conducted on feedback control based on HD streaming data to improve product quality, especially in the presence of incomplete responses. To address this challenge, this article proposes a novel tensor-based automatic control method for partially observed HD streaming data, which consists of two stages: offline modeling and online control. In the offline modeling stage, we propose a one-step approach integrating parameter estimation of the system model with missing value imputation for the response data. This approach (i) improves the accuracy of parameter estimation, and (ii) maintains a stable and superior imputation performance in a wider range of the rank or missing ratio for the data to be completed, compared to the existing data completion methods. In the online control stage, for each incoming sample, missing observations are imputed by balancing its low-rank information and the one-step-ahead prediction result based on the control action from the last time step. Then, the optimal control action is computed by minimizing a quadratic loss function on the sum of squared deviations from the target. Furthermore, we conduct two sets of simulations and one case study on semiconductor manufacturing to validate the superiority of the proposed framework. [ABSTRACT FROM AUTHOR]

Copyright of Dokuz Eylul University Muhendislik Faculty of Engineering Journal of Science & Engineering / Dokuz Eylül Üniversitesi Mühendislik Fakültesi Fen ve Mühendislik Dergisi is the property of Dokuz Eylul Universitesi Muhendislik Fakultesi Fen ve Muhendislik Dergisi and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we first study the equivalence between the third order tensor linear complementarity problem under the framework of t-product and the least squares problem under the t-product with nonnegative constraints, and based on their equivalence, apply the third order tensor linear complementarity problem to the t-product Arnoldi–Tikhonov regularization method for grayscale images deblurring. Secondly, we extend the definition of the third order tensor linear complementarity problem under the t-product to the order-p ( p > 3 ) tensor linear complementarity problem, propose a fixed point iterative method for solving the order-p ( p > 3 ) tensor linear complementarity problem, and prove that the equivalence between the third order tensor linear complementarity problem and the least squares problem under the t-product with nonnegative constraints also holds at the pth ( p > 3 ) order. Finally, we establish the tensor t-product model for color images deblurring with the within-channel and the cross-channel blurring, and propose the t-product Arnoldi–Tikhonov regularization method for this model. Moreover, we apply the fourth order tensor linear complementarity problem to solve the t-product Arnoldi–Tikhonov regularization method with nonnegative constraints. [ABSTRACT FROM AUTHOR]

Partially impaired sensor arrays pose a significant challenge in accurately estimating signal parameters. The occurrence of bad data is highly probable, resulting in random loss of source information and substantial performance degradation in parameter estimation. In this paper, a tensor variational sparse Bayesian learning (TVSBL) method is proposed for the estimate of direction of arrival (DOA) and polarization parameters jointly based on a conformal polarization sensitive array (CPSA), taking into account scenarios with the partially impaired sensor array. First, a sparse tensor-based received data model is developed for CPSAs that incorporates bad data. Then, a column vector detection method is proposed to diagnose the positions of the impaired sensors. In scenarios involving partially impaired sensor arrays, a low-rank matrix completion method is employed to recover the random loss of signal information. Finally, variational sparse Bayesian learning (VSBL) and minimum eigenvector methods are utilized sequentially to obtain the DOA and polarization parameters estimation, successively. Furthermore, the Cramér-Rao bound is given for the proposed method. Simulation results validated the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Let G be a connected general hypergraph of order n with rank r. The unique positive eigenvector x with -n i=1 x r i = 1 corresponding to the spectral radius ρ(G) is called the principal eigenvector of G. In this paper, the relation between each entry of the principal eigenvector of G and the vertex degree associated with this entry is presented. And some bounds for the extreme entries of the principal eigenvector are obtained. As applications, we give some bounds of the spectral radius of G. [ABSTRACT FROM AUTHOR]

Two approximation algorithms are proposed for ℓ 1 -regularized sparse rank-1 approximation to higher-order tensors. The algorithms are based on multilinear relaxation and sparsification, which are easily implemented and well scalable. In particular, the second one scales linearly with the size of the input tensor. Based on a careful estimation of the ℓ 1 -regularized sparsification, theoretical approximation lower bounds are derived. Our theoretical results also suggest an explicit way of choosing the regularization parameters. Numerical examples are provided to verify the proposed algorithms. [ABSTRACT FROM AUTHOR]

The need to know a few singular triplets associated with the largest singular values of a third‐order tensor arises in data compression and extraction. This paper describes a new method for their computation using the t‐product. Methods for determining a couple of singular triplets associated with the smallest singular values also are presented. The proposed methods generalize available restarted Lanczos bidiagonalization methods for computing a few of the largest or smallest singular triplets of a matrix. The methods of this paper use Ritz and harmonic Ritz lateral slices to determine accurate approximations of the largest and smallest singular triplets, respectively. Computed examples show applications to data compression and face recognition. [ABSTRACT FROM AUTHOR]

The Black Sea, situated between Türkiye, Bulgaria, Romania, Ukraine, and Russia, is tectonically separated into two different sub-basins: Eastern and Western Black Sea. These two sub-basins have been a target of interest for oil and gas exploration for several decades. Although the participation of the Black Sea Basin in the global oil market is very small compared to the Caspian Sea, this basin is considered a potential hydrocarbon deposit since both areas have similar characteristics in terms of source rock. In this study, satellite-derived Bouguer and free-air gravity data were interpreted to disclose the prospective hydrocarbon reservoirs and gas hydrate deposits within the Western Black Sea Basin. The locations of the maxima identified in the I2 invariants map were assessed as five substantial hydrocarbon prospective zones three of which are in the Turkish Exclusive Economic Zone. Numerous oil and gas seeps are evidence of lateral and vertical hydrocarbon migration from the source rock through major faults in the WBSB where the maximum I2 anomalies are observed. [ABSTRACT FROM AUTHOR]

Attention deficit hyperactivity disorder (ADHD) is a neurological disorder that affects an individual's behavior. The rising cases of ADHD among children and adolescents worldwide have raised the concern and require techniques for its early diagnosis and identification. The symptoms of ADHD are characterized by patterns of hyperactivity, inattention, and impulsivity. Recent advances in neuroimaging have allowed researchers to obtain the functional and structural patterns of the brain affected by ADHD. This work considers the resting state functional magnetic imaging (rs-fMRI) data and analyzes the functional connectivity of 40 subjects (20 ADHD and 20 healthy controls) through voxel size blood-oxygen-level-dependent (BOLD) signal. These BOLD signals are functionally relevant to the corresponding resting state networks (RSN). In this paper, we have proposed a modified deep learning-based bidirectional long short-term memory (BLSTM) model that automates the classification of ADHD through the identified voxels within the active region of the RSN. Initially, we have visualized the 28 active regions of RSN and time series of behavioral data of 40 subjects with 176 time stamps. Then, the proposed modified BLSTM has been trained by using the feature vector (40 × 261 × 28) for each subject and Adam hyper-parameter for optimization. The experimental results represent that the proposed model outperforms the many other models by achieving the classification accuracy of 87.50 % . We have also provided a detailed comparative analysis of the proposed model with the different existing state-of-the-art approaches. [ABSTRACT FROM AUTHOR]

In this paper, we consider a general tensor complementarity problem with interval parame- ters, and study the conditions under which, the existence and uniqueness of the solution of the problem are guaranteed. Furthermore, we proved that the solution set of the interval tensor complementarity problem is not necessarily convex. [ABSTRACT FROM AUTHOR]

Abstract Eddy viscosity is employed throughout the majority of numerical fluid dynamical models, and has been the subject of a vigorous body of research spanning a variety of disciplines. It has long been recognized that the proper description of eddy viscosity uses tensor mathematics, but in practice it is almost always employed as a scalar due to uncertainty about how to constrain the extra degrees of freedom and physical properties of its tensorial form. This manuscript borrows techniques from outside the realm of geophysical fluid dynamics to consider the eddy viscosity tensor using its eigenvalues and eigenvectors, establishing a new framework by which tensorial eddy viscosity can be tested. This is made possible by a careful analysis of an operation called tensor unrolling, which casts the eigenvalue problem for a fourth‐order tensor into a more familiar matrix‐vector form, whereby it becomes far easier to understand and manipulate. New constraints are established for the eddy viscosity coefficients that are guaranteed to result in energy dissipation, backscatter, or a combination of both. Finally, a testing protocol is developed by which tensorial eddy viscosity can be systematically evaluated across a wide range of fluid regimes.

Abstract The Lorentz–Dirac equation is formulated and studied in flat Minkowski spacetime. A concise, novel derivation of the equation is presented. The problem is then enlarged to study radiation damping of an electron moving through a gravitational field. The equation of motion is obtained for this case as well. It is suggested the study of the problem might motivate experiments which could shed light on the recent work related to the emergence of space-time and its structure by means of quantum effects such as quantum entanglement.

Multiband synthetic aperture radar (SAR) image fusion combines the scattering characteristics of targets from different bands to provide a comprehensive and informative output. However, the conventional SAR denoise and fusion separation framework often encounters the loss of fine details. This problem arises due to the inherent conflict between noise filtering and edge preservation. To address this issue, this article proposes the joint tensor-based low-rank total variation (TLRTV) dual-band SAR image denoise–fusion strategy. The proposed strategy formulates the denoise–fusion problem by integrating the LR and TV models. Furthermore, this problem is extended to a high-dimensional form using tensor representation. To effectively solve the TLRTV problem, an optimization method is developed based on the alternating direction method of multipliers. This method decomposes the TLRTV problem into a series of subproblems, allowing for an efficient and accurate solution. To evaluate the performance of the proposed TLRTV method, the real measurement dual-band SAR images obtained from our developed airborne multiband SAR system are utilized to compare with other existing denoise and fusion separation methods. The extensive experimental results demonstrate the superiority of the TLRTV approach achieving better fusion results, particularly in the presence of noise interference.

With the evolution of the sixth generation (6G) mobile communication technology, the terahertz (THz) spectrum has attracted much attention in wireless communication applications due to its high bandwidth and low signal transmission delay. However, the introduction of the THz spectrum leads to higher path transmission losses and complex signal attenuation. This makes user sensing and localization in THz massive multiple-input multiple-output (MIMO) systems more challenging. In this paper, we investigate the user sensing and localization problem in THz massive MIMO systems with the assistance of reconfigurable intelligent surfaces (RIS). Firstly, the received signal is modeled as a tensor, and a parallel factor (PARAFAC) method is proposed. The minimum description length (MDL) is then utilized to detect the number of scattering paths in the channel. On this basis, the alternating least squares (ALS) algorithm is employed to estimate the factor matrices, followed by the utilization of a straightforward correlation-based approach to obtain channel parameter information. Finally, the positions of users and scattering points are estimated based on the geometric relationship between channel parameters and location coordinates. The simulation results have verified the effectiveness of the proposed scheme compared to the existing competitive algorithms, and indicate that the proposed scheme exhibits superior parameter estimation performance and can achieve localization accuracy at the decimeter level.

The paper presents an iterative scheme called the general tensor regular splitting iterative (GTRS) method for solving the multilinear PageRank problem, which is based on a (weak) regular splitting technique and further accelerates the iterative process by introducing a parameter. The method yields familiar iterative schemes through the use of specific splitting strategies, including fixed-point, inner-outer, Jacobi, Gauss-Seidel and successive overrelaxation methods. The paper analyzes the convergence of these solvers in detail. Numerical results are provided to demonstrate the effectiveness of the proposed method in solving the multilinear PageRank problem.