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A real symmetric tensor is completely positive (CP) if it is a sum of symmetric tensor powers of nonnegative vectors. We propose a dehomogenization approach for studying CP tensors. This gives new Moment-SOS relaxations for CP tensors. Detection for CP tensors and the linear conic optimization with CP tensor cones can be solved more efficiently by the dehomogenization approach. [ABSTRACT FROM AUTHOR]

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]

Multi view clustering methods have become a research hotspot with the increasing diversity of data acquisition techniques. However, most clustering methods underestimate the impact of noise and complementary structural information of the data. Moreover, they often ignore the reverse guidance of clustering results on the optimization process of low rank tensors. To address these issues, this paper proposed a multi-view clustering method based on structured tensor learning (MCSTL). First, it further denoised the initial representation tensor to enhance its accuracy and robustness. At the same time, it complementarily learnt local structure, global structure, and high-order correlation across different views, which improved the consistency between the representation tensor and the intrinsic cluster structure of the original data. Then, it learnt a unified feature matrix from the affinity matrix of cross-view information fusion, and utilized the implicit clustering structure information within it to inversely guide the optimization process of the representation tensor. Lastly, it imposed an orthogonal constraint on the feature matrix, which provided soft label information of the data and allows for a direct clustering interpretation of the model. The experimental results show the MCSTL performs well in all six clustering evaluation metrics, with 27 out of 30 metrics reaching the optimal level, fully verifying the effectiveness and superiority of the MCSTL method. [ABSTRACT FROM AUTHOR]

We prove a lower bound on the dimension of the set of maximal border subrank tensors. This is the first such bound of its type. [ABSTRACT FROM AUTHOR]

This paper presents an Enhanced Multilinear Principal Component Analysis (EMPCA) algorithm, an improved variant of the traditional Multilinear Principal Component Analysis (MPCA) tailored for efficient dimensionality reduction in high-dimensional data, particularly in image analysis tasks. EMPCA integrates random singular value decomposition to reduce computational complexity while maintaining data integrity. Additionally, it innovatively combines the dimensionality reduction method with the Mask R-CNN algorithm, enhancing the accuracy of image segmentation. Leveraging tensors, EMPCA achieves dimensionality reduction that specifically benefits image classification, face recognition, and image segmentation. The experimental results demonstrate a 17.7% reduction in computation time compared to conventional methods, without compromising accuracy. In image classification and face recognition experiments, EMPCA significantly enhances classifier efficiency, achieving comparable or superior accuracy to algorithms such as Support Vector Machines (SVMs). Additionally, EMPCA preprocessing exploits latent information within tensor structures, leading to improved segmentation performance. The proposed EMPCA algorithm holds promise for reducing image analysis runtimes and advancing rapid image processing techniques. [ABSTRACT FROM AUTHOR]

Multilinear PageRank is a variant of the PageRank algorithm that takes into account multiple relationships among nodes in a network. This algorithm can make web page ranking more efficient and accurate by considering multiple types of connections at once. The higher-order power method is commonly used to calculate the multilinear PageRank vector due to its ease of implementation and low storage needs, and because it is a natural extension of the traditional power method used in the PageRank algorithm. However, the convergence of this method is not guaranteed, and even when it occurs, the process is often slow. In this paper, we show how some vector extrapolation methods such as minimal polynomial extrapolation (MPE) and reduced rank extrapolation (RRE) could be used for accelerating the computation of the fixed-point multilinear PageRank. [ABSTRACT FROM AUTHOR]

Multilinear PageRank is a variant of the well-known PageRank model. With this model, web page ranking can be more accurate and efficient by taking into account higher-order connections between pages. The higher-order power method is commonly employed for computing the multilinear PageRank vector, since it is a natural extension of the traditional power method used in the PageRank algorithm. However, this method may not be efficient when the hyperlink tensor becomes large or the damping factor value fails to meet the necessary conditions for convergence. In this work, we propose a novel approach to efficiently computing the multilinear PageRank vector using tensor splitting and vector extrapolation methods. [ABSTRACT FROM AUTHOR]

Many real-world datasets are represented as tensors, i.e., multi-dimensional arrays of numerical values. Storing them without compression often requires substantial space, which grows exponentially with the order. While many tensor compression algorithms are available, many of them rely on strong data assumptions regarding its order, sparsity, rank, and smoothness. In this work, we propose TensorCodec, a lossy compression algorithm for general tensors that do not necessarily adhere to strong input data assumptions.TensorCodec incorporates three key ideas. The first idea is neural tensor-train decomposition (NTTD) where we integrate a recurrent neural network into Tensor-Train Decomposition to enhance its expressive power and alleviate the limitations imposed by the low-rank assumption. Another idea is to fold the input tensor into a higher-order tensor to reduce the space required by NTTD. Finally, the mode indices of the input tensor are reordered to reveal patterns that can be exploited by NTTD for improved approximation. In addition, we extend TensorCodec to enable the lossy compression of tensors with missing entries, often found in real-world datasets. Our analysis and experiments on 8 real-world datasets demonstrate that TensorCodec is (a) Concise: it gives up to 7.38 × more compact compression than the best competitor with similar reconstruction error, (b) Accurate: given the same budget for compressed size, it yields up to 3.33 × more accurate reconstruction than the best competitor, (c) Scalable: Its empirical compression time is linear in the number of tensor entries, and it reconstructs each entry in logarithmic time. Our code and datasets are available at https://github.com/kbrother/TensorCodec. [ABSTRACT FROM AUTHOR]

Avatar is one of the most intuitive central components in Metaverse and faces serious security problems, particularly during the interaction with each other. In this article, we consider the problem of timely detecting the stealthy anomaly in the avatar interaction, which is crucial for security and privacy in Metaverse. With this goal, a new tensor summary statistic is proposed first to well depict the statistical discrepancy between normal and anomalous interaction volume samples, even when anomalies are stealthy. The proposed tensor summary statistic is established from the tensor linear representation residual, which naturally implies the statistical probability that an interaction volume sample lies within or deviates from the tensor lateral space. Moreover, a convex optimization programme is introduced to robustly recover the tensor lateral space in the presence of anomalous samples, thereby enhancing the robustness of our tensor summary statistic. On the basis of the tensor summary statistic, a non-parametric statistic framework is developed for the real-time detection of the stealthy interaction volume anomaly. We also provide theoretical analysis concerning its detection performance and parameter selection. Extensive experiments using synthetic and real-world datasets verify our effectiveness and superiority. Compared with benchmark methods, the proposed detection scheme achieves significantly lower detection delay and higher false alarm period, particularly in the detection of stealthy anomalies with a low change rate. [ABSTRACT FROM AUTHOR]

Data loss is a common problem in intelligent transportation systems (ITSs). And the tensor‐based interpolation algorithm has obvious superiority in multidimensional data interpolation. In this paper, a Bayesian robust tensor decomposition method (MBRTF) based on the Markov chain Monte Carlo (MCMC) algorithm is proposed. The underlying low CANDECOMP/PARAFAC (CP) rank tensor captures the global information, and the sparse tensor captures local information (also regarded as anomalous data), which achieves a reliable prediction of missing terms. The low CP rank tensor is modeled by linear interrelationships among multiple latent factors, and the sparsity of the columns on the latent factors is achieved through a hierarchical prior approach, while the sparse tensor is modeled by a hierarchical view of the Student‐t distribution. It is a challenge for traditional tensor‐based interpolation methods to maintain a stable performance under different missing rates and nonrandom missing (NM) scenarios. The MBRTF algorithm is an effective multiple interpolation algorithm that not only derives unbiased point estimates but also provides a robust method for the uncertainty measures of these missing values. [ABSTRACT FROM AUTHOR]

In this paper, we derive distributional convergence rates for the magnetization vector and the maximum pseudolikelihood estimator of the inverse temperature parameter in the tensor Curie–Weiss Potts model. Limit theorems for the magnetization vector have been derived recently in Bhowal and Mukherjee (arXiv preprint, arXiv:2307.01052, 2023), where several phase transition phenomena in terms of the scaling of the (centered) magnetization and its asymptotic distribution were established, depending upon the position of the true parameters in the parameter space. In the current work, we establish Berry–Esseen type results for the magnetization vector, specifying its rate of convergence at these different phases. At "most" points in the parameter space, this rate is N - 1 / 2 (N being the size of the Curie–Weiss network), while at some special points, the rate is either N - 1 / 4 or N - 1 / 6 , depending upon the behavior of the fourth derivative of a certain negative free energy function at these special points. These results are then used to derive Berry–Esseen type bounds for the maximum pseudolikelihood estimator of the inverse temperature parameter whenever it lies above a certain criticality threshold. [ABSTRACT FROM AUTHOR]

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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 study the generalized mixed variational inequality, which encompasses both the generalized variational inequality and the mixed variational inequality. The core contribution of this paper is twofold. Firstly, by utilizing the principles of degree theory, we establish certain sufficient conditions for the existence of solutions to the generalized mixed variational inequality. Additionally, we formulate a sufficient condition that ensures the uniqueness of these solutions. Secondly, we recognize that the conditions outlined in our theorem are inapplicable to the generalized mixed polynomial variational inequality, a subclass within the broader family of generalized mixed variational inequalities. To address this, we employ an exceptional family of elements and establish an existence and uniqueness theorem specifically tailored for the generalized mixed polynomial variational inequality. [ABSTRACT FROM AUTHOR]

Abstract Multimodal brain network analysis enables a comprehensive understanding of neurological disorders by integrating information from multiple neuroimaging modalities. However, existing methods often struggle to effectively model the complex structures of multimodal brain networks. In this paper, we propose a novel tensor-based graph convolutional network (TGNet) framework that combines tensor decomposition with multi-layer GCNs to capture both the homogeneity and intricate graph structures of multimodal brain networks. We evaluate TGNet on four datasets—HIV, Bipolar Disorder (BP), and Parkinson’s Disease (PPMI), Alzheimer’s Disease (ADNI)—demonstrating that it significantly outperforms existing methods for disease classification tasks, particularly in scenarios with limited sample sizes. The robustness and effectiveness of TGNet highlight its potential for advancing multimodal brain network analysis. The code is available at https://github.com/rongzhou7/TGNet .

AbstractStructured high-dimensional streaming data offers abundant information that is crucial for process feedback control. Nevertheless, traditional control models predominantly emphasize the global patterns of spatiotemporal correlation within responses, often neglecting the local correlation structure. This oversight can be problematic in applications where local correlations play a significant role, such as temperature control of composite plates. Additionally, these models typically fail to incorporate local patterns within the spatial influence of control variables, an essential aspect considering the location-sensitive nature of control impact. Moreover, in practice, the suboptimal uniform placement of control variables can significantly impact the effectiveness of control strategies under insufficient control resources. To address these issues, we propose a tensor-based feedback control model for locally structured high-dimensional streaming data under limited control capabilities. For system modeling, we employ kernel distributions to capture the local structure within (i) the response autocorrelation and (ii) the spatial impact of location-sensitive control variables. For online control, we develop a dynamic control strategy to optimize controller placement, enhancing control efficiency despite resource constraints. Finally, we validate the effectiveness of our proposed framework through simulations and a case study. [ABSTRACT FROM AUTHOR]

Multimodal brain network analysis enables a comprehensive understanding of neurological disorders by integrating information from multiple neuroimaging modalities. However, existing methods often struggle to effectively model the complex structures of multimodal brain networks. In this paper, we propose a novel tensor-based graph convolutional network (TGNet) framework that combines tensor decomposition with multi-layer GCNs to capture both the homogeneity and intricate graph structures of multimodal brain networks. We evaluate TGNet on four datasets—HIV, Bipolar Disorder (BP), and Parkinson's Disease (PPMI), Alzheimer's Disease (ADNI)—demonstrating that it significantly outperforms existing methods for disease classification tasks, particularly in scenarios with limited sample sizes. The robustness and effectiveness of TGNet highlight its potential for advancing multimodal brain network analysis. The code is available at https://github.com/rongzhou7/TGNet. [ABSTRACT FROM AUTHOR]

We study spontaneous the spin polarization of quark matter with flavor S U (2) symmetry at zero temperature in the NJL model. In a relativistic framework, there are two types of spin–spin interactions: axial vector (AV) and tensor (T), which accordingly give rise to different types of spin-polarized materials. When the spin–spin interaction is sufficiently strong, the spin-polarized phase emerges within a specific density region. As the spin–spin interaction becomes stronger, this phase extends over a higher-density region beyond the critical density of chiral restoration in normal quark matter. We show that the spin-polarized phase leads to another kind of spontaneous chiral symmetry breaking phase. [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]

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]

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]

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]

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.

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.

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]

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]

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]

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]

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]

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]

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]

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