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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 .

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]

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.

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]

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 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.

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.

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]

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.

Utilizing unmanned aerial vehicle (UAV) technology in communication holds promise for meeting the increasing data rate demands in future wireless systems due to its flexibility. Meanwhile, reconfigurable intelligent surface (RIS) has garnered increased attention for their potential to enhance wireless communication performance through intelligent control of the transmission environment. In this paper, we first combine the UAV and the RISs to construct an Internet of Things (IoT) uplink transmission system, where the UAV serves as an aerial relay to collect data from IoT terminal (IT) and forward it to base stations (BS), while RISs assist communication to reduce congestion. Then, a parallel factor (PARAFAC) tensor model is formulated at the BS. At last, the iterative alternating least squares (ALS) algorithm and the closed-form singular value decomposition (SVD) algorithm are derived to fit the constructed tensor model for joint channel estimation and symbol detection. Compared with the competitive algorithms, the two proposed algorithms offer lower computational complexity and superior channel estimation performance. Furthermore, the proposed algorithms exhibit good symbol detection capabilities even at higher transmission rates. The numerical results demonstrate the effectiveness of the proposed algorithms.

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.

To achieve a green and sustainable wireless communication network, the reconfigurable intelligent surface (RIS) technology has emerged as an emerging technology. The RIS-assisted mmWave massive MIMO vehicle-to-everything (V2X) communications can support enhanced V2X applications for connected and automated vehicles. The design of RIS-assisted mmWave V2X communications is though not exempt of challenges as a result of fast time-varying propagation and highly dynamic vehicular networks and topologies. To address some of these challenges, we propose an attention-deep reinforcement learning jointly beamforming based on tensor decomposition for RIS-assisted mmWave massive MIMO system to improving the safety and traffic efficiency of cooperative automated driving. The received signal of the vehicle is decomposed into four-dimensional tensors and study the channel, frequency and time attention (CFTA) algorithm using the low order tensor characteristics of mmWave channels. This algorithm can extract the frequency and time characteristics of the channel, and obtain the millimeter wave characteristic channel between the corresponding base station (BS) and the vehicles. The problem of jointly optimization of beamforming matrix at the BS and the phase shift matrix on the RIS is constructed and solved using deep reinforcement learning (DRL). Unlike traditional alternating optimization methods, the integration of DRL techniques into the optimal design of the system enables the observation of immediate rewards, learning from the environment and improving it to obtain optimal solutions to high-dimensional jointly beamforming optimization problems ultimately. The simulation results show that the proposed algorithm has meaningful performance compared with other methods. [ABSTRACT FROM AUTHOR]

Denoising of time series is still a challenge, especially when the spectral components of wanted signal and noise overlap. However, denoising is an inverse problem and its solution is of often ambiguous. Standard methods often use predefined functions, e.g., sine waves, to decompose a time series into wanted and unwanted parts. More recent methods calculate the basis for the representation of a time series by the time series itself. Such a method is singular spectrum analysis (SSA). SSA uses Hankel matrix embedding of a time series and singular value decomposition to determine wanted and unwanted components, e.g., noise. The wanted components are then used to compute the wanted part of the time series. Here we present a method that provides an extension of SSA for analyzing and - in particular - denoising multi-channel time series, i.e., multi-channel SSA-based denoising of multi-channel time series (MSSAD). The performance of the new method, which is based on Hankel embedding and tensor decomposition, is demonstrated on a 12-lead ECG.

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 Background It is important to differentiate cervical lymph nodes. So, this study aims to assess the ability of diffusion tensor imaging (DTI) in differentiating cervical lymphadenopathy (LNs). Materials and methods This retrospective study was done upon 100 patients with cervical LNs who had DTI over a year period. The fractional anisotropy (FA) and the mean diffusivity (MD) values of LNs were measured. Results This study was done upon 100 patients (the mean age 45 ± 2 years (standard deviation [SD]), 63 men). The mean MD and FA of the malignant LNs (0.83 ± 0.14 × 10−3 mm2/s, 0.26 ± 0.07) were significantly different; (P = 0.001) than those of benign LNs (1.32 ± 0.33 × 10–3 mm2/s, 0.22 ± 0.09). MD of 0.94 × 10–3 mm2/s and FA of 0.21 were used to discriminate malignant and benign LNs, AUC 0.892 and 0.758, and 84% and 71% accuracy, respectively. Combined parameters revealed AUC of 0.914 and 81%. The mean MD and FA of the metastatic LNs (0.86 ± 0.12 × 10–3 mm2/s, 0.25 ± 0.07) were statistically different; (P = 0.001, 0.03) than those of lymphomatous nodes (0.66 ± 0.13 × 10–3 mm2/s, 0.28 ± 0.02). The AUC of the MD and FA used to distinguish metastatic from lymphomatous nodes was 0.82, 0.711, (0.71 × 10–3 mm2/s, 0.27) cutoff values, and 95.3%, 73.4% accuracy, respectively. Combined parameters revealed 0.824 AUC, 95.3% accuracy, 98.2% sensitivity, and 75% specificity. There was a significant statistical difference in MD between well-moderately (P = 0.001) versus poorly differentiated metastatic LNs and stages I and II (P = 0.018) versus stages III and IV of metastatic cervical LNs. Conclusions Combining FA and MD is a promising technique that can play a major role in distinguishing different categories of cervical LNs.

Abstract To achieve a green and sustainable wireless communication network, the reconfigurable intelligent surface (RIS) technology has emerged as an emerging technology. The RIS-assisted mmWave massive MIMO vehicle-to-everything (V2X) communications can support enhanced V2X applications for connected and automated vehicles. The design of RIS-assisted mmWave V2X communications is though not exempt of challenges as a result of fast time-varying propagation and highly dynamic vehicular networks and topologies. To address some of these challenges, we propose an attention-deep reinforcement learning jointly beamforming based on tensor decomposition for RIS-assisted mmWave massive MIMO system to improving the safety and traffic efficiency of cooperative automated driving. The received signal of the vehicle is decomposed into four-dimensional tensors and study the channel, frequency and time attention (CFTA) algorithm using the low order tensor characteristics of mmWave channels. This algorithm can extract the frequency and time characteristics of the channel, and obtain the millimeter wave characteristic channel between the corresponding base station (BS) and the vehicles. The problem of jointly optimization of beamforming matrix at the BS and the phase shift matrix on the RIS is constructed and solved using deep reinforcement learning (DRL). Unlike traditional alternating optimization methods, the integration of DRL techniques into the optimal design of the system enables the observation of immediate rewards, learning from the environment and improving it to obtain optimal solutions to high-dimensional jointly beamforming optimization problems ultimately. The simulation results show that the proposed algorithm has meaningful performance compared with other methods.

The study develops and describes, in general, new models and devices of energy converters. The proposed devices operate on the basis of a rectangular parallelepiped with length a, height b, and width c ( ) made of anisotropic material. The selected crystallographic axes of the mentioned material are located in the side face (a×b) of a rectangular plate, and one of the crystallographic axes is inclined at a certain angle γ to length a. The method of energy transformation is based on the effect of polarization of the volume of such an anisotropic plate and the appearance as a longitudinal one , and transverse components of the vortex energy field. In the first case, a unipolar anisotropic material characterized by a second-rank tensor is used to make anisotropic platinum . All tensor coefficients are positive . When the proper boundary conditions are met, vortices with a laminar flow character are formed in the volume of a unipolar anisotropic medium. In the case of counter-electric E, magnetic H, gravitational G fields and flows, electric current I and heat flow Q, an energy conversion coefficient of which does not exceed 1 (m ≤ 1) is observed. For the case of bipolar anisotropic materials, when the boundary conditions are met, energy vortices are formed, characterized by a turbulent flow pattern. Such a bipolar anisotropic material is described by a second-rank tensor . We consider the case when one of the coefficients of the tensor is negative . Turbulent energy vortices, interacting with the external environment, make it possible to obtain a conversion factor significantly greater than 1 ( ). In general, this significantly expands the existing capabilities and leads to the emergence of new methods and energy conversion devices created on them.

The vibrational spectra of polyatomic C4H8O2 dioxane molecules which is symmetric in the polarizability tensor have been studied by the method of Raman scattering in a low-frequency range (0–1400 cm−1). The possibilities of their use for obtaining information about the features of interactions of molecules, their are shown in vibrational and orientational motion dynamics. For the dioxan molecule, 8 spectra associated with vibrations of C-H atoms were observed and it was found that their vibration frequencies correspond to approximately 400–1300 cm−1. The values of the relaxation parameters characterizing the laws of interaction of atoms related to hydrogen bonding were found. Manifestation of these spectra was analyzed on the basis of the change in the polarizability of the molecule

Unmanned aerial vehicles (UAVs) can support low-cost, highly mobile communications while making it possible to establish dedicated terrestrial networks. To overcome the pointing error (PE) and beam misalignment of millimeter-wave large-scale multiple-input multiple-output/free-space optics (MIMO/FSO) caused by UAV jitter, a millimeter-wave massive MIMO/FSO hybrid beamforming method based on tensor train decomposition is proposed. This approach is used for reconfigurable intelligent surface (RIS) network-assisted UAV millimeter-wave massive MIMO/FSO to improve system spectral efficiency. Firstly, the high-dimensional channel of RIS-assisted millimeter-wave massive MIMO/FSO in UAV is represented as a low-dimensional channel by tensor training decomposition. Secondly, the two-way gated recursive unit attention neural network model can effectively solve the FSO PE caused by UAV jitter, and the fast fading channel and Doppler frequency shift are estimated by the Fast Cyclic Tensor Power Method (FCTPM) based on tensor training decomposition. Finally, the RIS phase shift matrix is optimized by singular value decomposition. The hybrid beamforming and RIS phase-shift matrices were estimated by using the low-complexity phase extraction alternating minimization method to solve the beam misalignment problem. Simulation results show that by using the proposed method, the spectrum utilization rate is improved by 23.6% compared with other methods. [ABSTRACT FROM AUTHOR]

In the present paper, we study two different approaches of tensor decomposition. The first part aims to study some properties of tensors that result from the fact that some components are vanishing in certain coordinates. It is proven that these conditions allow tensor decomposition, especially (1, σ), σ = 1 , 2 , 3 tensors. We apply the results for special tensors such as the Riemann, Ricci, Einstein, and Weyl tensors and the deformation tensors of affine connections. Thereby, we find new criteria for the Einstein spaces, spaces of constant curvature, and projective and conformal flat spaces. Further, the proof of the theorem of Mikeš and Moldobayev is repaired. It has been used in many works and it is a generalization of the criteria formulated by Schouten and Struik. The second part deals with the properties of a special differential operator with respect to the general decomposition of tensor fields on manifolds with affine connection. It is shown that the properties of special differential operators are transferred to the components of a given decomposition. [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.

Background It is important to diferentiate cervical lymph nodes. So, this study aims to assess the ability of difusion tensor imaging (DTI) in diferentiating cervical lymphadenopathy (LNs). Materials and methods This retrospective study was done upon 100 patients with cervical LNs who had DTI over a year period. The fractional anisotropy (FA) and the mean difusivity (MD) values of LNs were measured. Results This study was done upon 100 patients (the mean age 45 ± 2 years (standard deviation [SD]), 63 men). The mean MD and FA of the malignant LNs (0.83 ± 0.14 × 10−3 mm²/s, 0.26 ± 0.07) were significantly different; (P = 0.001) than those of benign LNs (1.32 ± 0.33 × 10–3 mm²/s, 0.22 ± 0.09). MD of 0.94 × 10–3 mm²/s and FA of 0.21 were used to discriminate malignant and benign LNs, AUC 0.892 and 0.758, and 84% and 71% accuracy, respectively. Combined parameters revealed AUC of 0.914 and 81%. The mean MD and FA of the metastatic LNs (0.86 ± 0.12 × 10–3 mm²/s, 0.25 ± 0.07) were statistically different; (P = 0.001, 0.03) than those of lymphomatous nodes (0.66 ± 0.13 × 10–3 mm²/s, 0.28 ± 0.02). The AUC of the MD and FA used to distinguish metastatic from lymphomatous nodes was 0.82, 0.711, (0.71 × 10–3 mm²/s, 0.27) cutoff values, and 95.3%, 73.4% accuracy, respectively. Combined parameters revealed 0.824 AUC, 95.3% accuracy, 98.2% sensitivity, and 75% specificity. There was a significant statistical difference in MD between well-moderately (P = 0.001) versus poorly differentiated metastatic LNs and stages I and II (P = 0.018) versus stages III and IV of metastatic cervical LNs. Conclusions Combining FA and MD is a promising technique that can play a major role in distinguishing diferent categories of cervical LNs. [ABSTRACT FROM AUTHOR]

In order to mine the manifold information in different views, a hypergraph regularized low-rank tensor multi-view subspace clustering via L1 norm constraint (HLRT-MSCL1) is proposed. The tensor in this method is composed of subspace representation matrix of different views, which is used to mine the high order correlations between various views, and the low-rank restriction is performed for the tensor to effectively decrease the redundant data of the subspace representation. In addition, the HLRT-MSCL1 method considers the local geometric information of different views and imposes sparse constraint on the subspace representation matrix, so that its recognition performance is improved to a certain extent. Moreover, an effective method for solving the HLRT-MSCL1 model is also designed, and its convergence is discussed. The results of extensive experiments conducted on five challenging data sets show that the proposed method objectively achieves a highly competent performance compared with several other state-of-the-art multi-view clustering methods. [ABSTRACT FROM AUTHOR]

The recommender system's primary purpose is to estimate the user's desire and provide a list of items predicted from the appropriate information. Also, context-aware recommendation systems are becoming more and more favorite since they could provide more accurate or personalized recommendation information than traditional recommendation techniques. However, a context-aware recommendation system suffers from two fundamental limitations known as cold start and sparse data. Singular value decomposition has been successfully integrated with some traditional recommendation algorithms. However, the basic singular value decomposition can only extract the feature vectors of users and items, resulting in lower recommendation precision. To improve the recommendation performance and reduce the challenge of cold start and sparse data, we propose a new context-aware recommendation algorithm, named CSSVD. First, in the CSSVD matrix, using the IFPCC and DPCC similarity criteria, the item's user property attribute matrices are created, respectively, creating the SSVD matrix for the cold start problem. In the second step, through the CWP similarity criterion on the contextual information, the context matrix is created, which according to the SSVD matrix created in the previous step, creates a three-dimensional matrix based on tensor properties, providing the problem of sparse data. We have used the IMDB and STS data collection because of implementing user features, item features, and contextual data for analyzing the recommended method. Experiential results illustrate that the proposed algorithm CSSVD is better than TF, HOSVD, BPR, and CTLSVD in terms of Precision, Recall, F-score, and NDCG measure.Results show the improvement of the recommendations to users through alleviating cold start and sparse data. [ABSTRACT FROM AUTHOR]

In real life, only partial information of samples is available everywhere, this makes Incomplete multi-view clustering (IMVC) becomes a significant research topic to handle data loss situations. Recently, several methods leverage the anchor strategy by selecting fixed anchors to handle the challenging large-scale IMVC. However, all of them ignore the guidance of prior information hidden in the bipartite graph. Therefore, we propose a novel Anchor Pseudo-supervise Large-scale Incomplete Multi-view Clustering (AP-LIMC) method by introducing a prior indicator matrix as a pseudo-supervise anchor learning paradigm. Specifically, the prior indicator matrix is first introduced to control the distribution of anchors in each cluster. Then, an anchor pseudo-supervise learning framework is designed to generate high-quality anchors and a unified bipartite graph with prior indicator supervision. In addition, we design an optimized process with linear computational and extensive experiments on multiple public datasets with recent advances to validate the effectiveness, superiority, and efficiency. For example, on the Stl10 dataset, the performance of the proposed AP-LIMC improved by 23.95%,15.71%,27.39%, and 18.24% in terms of four evaluation metrics, respectively.

Any object can have many implementations in the form of digital images and any digital image can be processed many times increasing or decreasing accuracy and reliability. Digital colorimetry faces the need to work out issues of ensuring accuracy, metrological traceability and reliability. The purpose of this work was to generalize approaches to the description of multidimensional quantized spaces and show the possibilities of their adaptation to digital colorimetry. This approach will minimize the private and global risks in measurements.For color identification digital colorimetry uses standard color models and spaces. Most of them are empirical and are improved during the transition from standard to real observation conditions taking into account the phenomena of vision and the age of observers. From the point of view of measurement, a digital image can be represented by a combinatorial model of an information and measurement channel with the appearance of the phenomenon of a color covariance hypercube requiring a significant amount of memory for data storage and processing. The transition from the covariance hypercube to high-dimensional matrices and tensors of the first, second and higher ranks provides the prospect of optimizing the color parameters of a digital image by the criterion of information entropy.Tensor calculus provides opportunities for expanding the dynamic range in color measurements describing multidimensional vector fields and quantized spaces with indexing tensors and decomposing them into matrices of low orders.The proposed complex approach based on tensor calculus. According to this approach the color space is a set of directed vector fields undergoing sampling, quantization and coding operations. Also it is a dynamic open system exchanging information with the environment at a given level and to identify color with specified levels of accuracy, reliability, uncertainty and entropy.

The paper is devoted to problems concerning the tensors with constant components, hemitropic tensors and pseudotensors that are of interest from the point of view of micropolar continuum mechanics. The properties and coordinate representations of tensors and pseudotensors with constant components are discussed. Based on an unconventional definition of a hemitropic fourth-rank tensor, a coordinate representations in terms of Kronecker deltas and metric tensors are given. A comparison of an arbitrary hemitropic fourth-rank tensor and a tensor with constant components are discussed. The coordinate representations for constitutive tensors and pseudotensors used in mathematical modeling of linear hemitropic micropolar continuums are given in terms of the metric tensor.The covariant constancy of fourth-rank pseudotensors with constant components and hemitropic tensors is considered and discussed.

A linear transformation from vector space to another vector space can be represented as a matrix. This close relationship between the matrix and the linear transformation is helpful for the study of matrices. In this paper, the tensor is regarded as a generalization of the matrix from the viewpoint of the linear transformation instead of the quadratic form in matrix theory; we discuss some operations and present some definitions and theorems related to tensors. For example, we provide the definitions of the triangular form and the eigenvalue of a tensor, and the theorems of the tensor QR decomposition and the tensor singular value decomposition. Furthermore, we explain the significance of our definitions and their differences from existing definitions. [ABSTRACT FROM AUTHOR]

In this paper, to extract the manifold information from multi-view data and enhance the clustering performance of a multi-view learning method, the multi-view subspace enhanced representation of manifold regularization and low-rank tensor constraint (MSERMLRT) method is introduced. Our model uses a tensor to explore the correlation between views. The tensor is constrained with a low-rank, and the purpose of such processing is to reduce the redundant information of the learned subspace representation. This model also uses the manifold information from multi-view data and imposes a sparse constraint on the product of itself and the transpose of the subspace representation matrix to enhance the diagonal block structure of the subspace representation, thereby improving its clustering effect to a certain extent. We also designed a helpful method for solving the MSERMLRT model and analyzed the convergence of our approach both theoretically and experimentally. The clustering performance on certain challenging datasets indicate that the MSERMLRT model is superior to many other advanced multi-view clustering methods. [ABSTRACT FROM AUTHOR]

Through the more available acoustic information or the polarization information provided, vector sensor arrays outperform the scalar sensor arrays in accuracy of localization. However, the cost of a vector sensor array is higher than that of a scalar sensor array. To reduce the cost of a two-dimensional (2-D) vector sensor array, a hybrid T-shaped sensor array consisting of two orthogonal uniform linear arrays (ULAs) is proposed, where one ULA is composed of acoustic vector sensors and the other is composed of scalar sensors. By utilizing the cross-correlation tensor between the received signals from the two ULAs, two virtual uniform rectangular arrays (URAs) of acoustic vector sensors are obtained, and they can be combined into a larger URA. It is shown that a larger acoustic vector sensor URA with M2+1 degrees of freedom (DOFs) can be obtained from the specially designed T-shaped array with M acoustic vector sensors and 2M scalar sensors. Furthermore, by means of the proposed tensor model for the larger URA, the inter-sensor spacing can be allowed to exceed greatly a half-wavelength. Accordingly, the proposed method can achieve both a high DOF and a large array aperture. Simulation results show that the proposed method has a better performance in 2-D direction-of-arrival estimation than some existing methods under the same array cost. [ABSTRACT FROM AUTHOR]

Higher-order tensor canonical polyadic decomposition (CPD) with one or more of the latent factor matrices being columnwisely orthonormal has been well studied in recent years. However, most existing models penalize the noises, if occurring, by employing the least squares loss, which may be sensitive to non-Gaussian noise or outliers, leading to bias estimates of the latent factors. In this paper, we derive a robust orthogonal tensor CPD model with Cauchy loss, which is resistant to heavy-tailed noise such as the Cauchy noise, or outliers. By exploring the half-quadratic property of the model, we develop the so-called half-quadratic alternating direction method of multipliers (HQ-ADMM) to solve the model. Each subproblem involved in HQ-ADMM admits a closed-form solution. Thanks to some nice properties of the Cauchy loss, we show that the whole sequence generated by the algorithm globally converges to a stationary point of the problem under consideration. Numerical experiments on synthetic and real data demonstrate the effectiveness of the proposed model and algorithm. [ABSTRACT FROM AUTHOR]

Graph-based change-point detection methods are often applied due to their advantages for using high-dimensional data. Most applications focus on extracting effective information of objects while ignoring their main features. However, in some applications, one may be interested in detecting objects with different features, such as color. Therefore, we propose a general graph-based change-point detection method under the multi-way tensor framework, aimed at detecting objects with different features that change in the distribution of one or more slices. Furthermore, considering that recorded tensor sequences may be vulnerable to natural disturbances, such as lighting in images or videos, we propose an improved method incorporating histogram equalization techniques to improve detection efficiency. Finally, through simulations and real data analysis, we show that the proposed methods achieve higher efficiency in detecting change-points. [ABSTRACT FROM AUTHOR]

The peculiarities of magnetic field propagation in a rectangular plate made of anisotropic material by the coefficient of magnetic permeability are considered. The dependences of the longitudinal and transverse components of the magnetic field on the geometric dimensions of the plate and the angle of inclination of the selected main crystallographic axes are established. For the first time, the possibility of concentrating the magnitude of the magnetic field strength was shown and the corresponding optimization of its magnitude was performed. Models of anisotropic magnetic concentrator are presented, and also the method of its creation on the basis of artificial anisotropic, both classical, and metamagnetic materials is offered. The application of the proposed method leads to the emergence of new energy converters of autonomous clean energy that does not require conventional energy sources. The method of magnetic field concentration using artificial anisotropic classical materials improves the performance of electromagnetic and magnetoelectric measuring systems, but does not fully meet the requirements of the latest technological devices. Further increase of the magnetic field concentration coefficient is achieved by using artificial anisotropic metamagnetic materials, which are characterized by a negative value of magnetic permeability in one of the selected main crystallographic directions. The proposed methods of magnetic field concentration will be used in electrical engineering, radio engineering, metrology and other related fields.

The digital deep integration of cyber space, physical space and social space facilitates the formation of Cyber–Physical–Social Systems (CPSS). Knowledge empowers CPSS to be capable of solving complex tasks and providing intelligent services. Knowledge representation, fusion, and reasoning are complementary in the knowledge graph and worthy of continuous exploration and in-depth research. Since the concept of knowledge representation learning has been proposed, various models have been developed and demonstrated their superior ability in knowledge fusion and reasoning. In this paper, we investigate the progress and function of representation learning models adopted in knowledge fusion and reasoning, and extract their commonalities and characteristics, which is a perspective that has never been developed before, to provide new ideas for scholars and realize the mutual promotion of knowledge fusion and reasoning. This paper comprehensively reviews classic methods and investigates advanced and emerging works, which are published in authoritative journals or conferences. Furthermore, to overcome the limitations of existing methods, we propose an integrated knowledge representation learning and application framework for CPSS and a series of tensor-based knowledge fusion and reasoning models. Finally, four future directions are discussed in this paper, we analyze the challenges faced by them and suggest promising solutions. • Extensive coverage of emerging advances about knowledge representation learning. • Targeted investigations and new perspectives about knowledge fusion and reasoning. • Tensor-enabled representation learning framework and promising models for CPSS. • Insightful prospects on representation learning-based knowledge reasoning and fusion. [ABSTRACT FROM AUTHOR]