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We provide a procedure to construct entanglement-assisted Calderbank–Shor–Steane (CSS) codes over qudits from the parity check matrices of two classical codes over $\mathbb {F}_q$ , where $q=p^k$ , $p$ is prime, and $k$ is a positive integer. The construction procedure involves the proposed Euclidean Gram–Schmidt orthogonalization algorithm, followed by a procedure to extend the quantum operators to obtain stabilizers of the code. Using this construction, we provide a construction of entanglement-assisted tensor product codes from classical tensor product codes over $\mathbb {F}_q$ . We further show that a nonzero rate entanglement-assisted tensor product code can be obtained from a classical tensor product code whose component codes yield zero rate entanglement-assisted CSS codes. We view this result as the coding analog of superadditivity.

In this paper we consider the concept of $KM$ -fuzzy metric spaces and we introduce a novel concept of $KM$ -single valued neutrosophic metric graphs based on $KM$ -fuzzy metric spaces. Then we investigate the finite $KM$ -fuzzy metric spaces with respect to $KM$ -fuzzy metrics and we construct the $KM$ -fuzzy metric spaces on any given non-empty sets. We try to extend the concept of $KM$ -fuzzy metric spaces to a larger class of $KM$ -fuzzy metric spaces such as union and product of $KM$ -fuzzy metric spaces and in this regard we investigate the class of products of $KM$ -single valued neutrosophic metric graphs. In the final, we define some operations such as tensor product, Cartesian product, semi-strong product, strong product, union, semi-ring sum, suspension, and complement of $KM$ -single valued neutrosophic metric graphs.

Tento příspěvek popisuje novou geometrickou transformace pomocí tenzorového součinu. Geometrické transformace se široce používají v mnoha aplikacích, zejména v CAD/CAM systémy, systémy pro stavebnictví, počítače grafika, virtuální a rozšířená realita atd. Základní geometrické transformace, např. rotace, translace, změna měřítka atd. se obvykle aplikují na body reprezentované v homogenních souřadnicích This contribution describes new useful geometric transformations using the tensor product. The geometric transformations are used widely in many applications, especially in CAD/CAM systems, systems for Civil Engineering, computer graphics, virtual and augmented reality, etc. The basic geometric transformations, e.g. rotation, translation, scaling, etc., are usually applied on points represented in homogeneous coordinates.

A new serendipity function space for hexahedral H(curl)-conforming finite elements, the so-called mixed-order serendipity space, is proposed. In the case of arbitrary parallelepiped meshes, the resulting asymptotic rate of convergence of the H(curl)-norm error is exponential in the base of the mesh size and in terms of the finite element order. Compared to tensor product spaces, the number of unknowns is much smaller, whereas the rate of convergence remains the same. While the proposed serendipity space is not suitable for elements of general shape, it allows the construction of hierarchical basis functions that are compatible with some of the costly but versatile tensor product spaces. This property permits mixing different finite element spaces within one mesh, without affecting conformity.For the general curvilinear case, this paper introduces an iso-serendipity finite element for curvilinear hexahedra, which employs the proposed serendipity space and basis for representing the fields and H1 serendipity basis for representing the geometry. As the main contribution, this element achieves the same convergence rate for both curvilinear and parallelepiped meshes, by means of a special yet simple mesh refinement technique. In contrast to isogeometric methods, it only requires certain interpolation points on curvilinear boundaries rather than the entire geometry mapping. All results are supported by mathematical proofs and validated experimentally via numerical examples.

The notion of safety in safety-critical applications can be represented by the concept of critical observability. The traditional concept of critical observability refers to possibility of detecting if the current state of a nondeterministic finite state machine (NFSM) belongs to a so-called critical set for any input and any initial state. In practice, the input is always of finite length and we may only care about the state of an FSM at a given time step or the inverse problem that how to find all possible inputs moving the FSM to a critical state of interest. Thus, we propose two new concepts about critical observability, critical observability with finite input and critical observability at a given time step, to cope with these two cases. From the view of algebraic model resulting from the theory of semi-tensor product (STP), we propose corresponding algebraic criteria and algorithms. They are based on matrix operation, which means that they naturally avoid the design of critical observers and can be solved efficiently by computers.

In this paper, we propose a new method based on semi-tensor product of matrices for solving quaternion Generalized Lyapunov Equation ${A^H}X + XA + \sum\limits_{j = 1}^m {N_j^H} X{N_j} + {C^H}C = 0$. First, we define a real vector representation of quaternion matrix and study its properties. By using semi-tensor product of matrices and this real vector representation, we transform the problem of quaternion matrix equation into the corresponding real matrix equation. And then, by studying the relationship between the real vector representations of Hermitian matrix and their independent elements, we reduce the complexity of the original problem. We derive the expression of the least squares Hermitian solution for the quaternion Generalized Lyapunov Equation ${A^H}X + XA + \sum\limits_{j = 1}^m {N_j^H} X{N_j} + {C^H}C = 0$. All the expressions of solutions only involve real arithmetic. We propose the corresponding algorithms and give numerical experiments to illustrate the validity of our methods.

In this paper, a semi-tensor product (STP) approach is proposed to discuss the stability problem of a constrained switching system (CSS). The switching sequence is subject to the constraint of a nondeterministic finite automaton (NFA). By the mathematical tool, the constrained joint spectral radius (CJSR) of the CSS can be approximated. We propose the equivalence of a matrix-based expression of the CSS and its lifted l-dimensional switching system by using the STP approach. After that, the approximate relation between the constrained joint spectral radius (CJSR) /the constrained generalized spectral radius (CGSR) and the joint spectral radius (JSR)/ generalized spectral radius (GSR) is developed for the switching system. At the same time, there is a special case of NFA, where the equivalence relation between them also is proved.

This paper studies dynamics behaviors for a class of networked evolutionary games with weight by using semi-tensor product of matrices. Each player is able to choose some of the associated players as their own neighborhoods in weighted networked evolutionary games. Firstly, the model is put forward named networked evolutionary games with weight, which quantify the connections between players. Each player choose their neighborhoods who meet their own mental standards from connected people. Secondly, an algebraic form of the model is got via the semi-tensor product. The dynamical behavior of evolutionary networked games is discussed through it, then some interesting results about attractors are obtained. Finally, an example is showed to illustrate the theoretical results in this paper and explains that the new results can work well.

Image restoration with corruptions from combined multiple types of distortion is a challenging and practical problem. Recent studies show that a promising technique is to perform parallel “operations” to handle different types of distortion, which can be modeled by a deep neural network framework. However, the reconstruction may be dominated by a small number of operations due to the heterogeneous features generated by different operations. To handle this issue, we introduce a tensor 1×1 convolutional layer by leveraging high-order tensor fusion, which can not only harmonize the heterogeneous features but also take high order statistical information into account. To efficiently learn the large-scale kernel tensor resulted from the tensor product, we employ tensor network to represent kernels, which is able to convert the exponential growth of the dimension to linear growth. Armed with this new layer, we propose high-order operation-wise attention network for the task of multi-distorted image restoration. The experimental results demonstrated that the proposed method outperforms the method with vanilla 1 × 1 convolutional layer in several typical tasks and is promising for more difficult tasks. Code is available at https://github.com/ZihaoH/High-order-OWAN.

We study the discrimination of a pair of orthogonal quantum states in the many-copy setting. This is not a problem when arbitrary quantum measurements are allowed, as then the states can be distinguished perfectly even with one copy. However, it becomes highly nontrivial when we consider states of a multipartite system and locality constraints are imposed. We hence focus on the restricted families of measurements such as local operation and classical communication (LOCC), separable operations (SEP), and the positive-partial-transpose operations (PPT) in this paper. We first study asymptotic discrimination of an arbitrary multipartite entangled pure state against its orthogonal complement using LOCC/SEP/PPT measurements. We prove that the incurred optimal average error probability always decays exponentially in the number of copies, by proving upper and lower bounds on the exponent. In the special case of discriminating a maximally entangled state against its orthogonal complement, we determine the explicit expression for the optimal average error probability and the optimal trade-off between the type-I and type-II errors, thus establishing the associated Chernoff, Stein, Hoeffding, and the strong converse exponents. Our technique is based on the idea of using PPT operations to approximate LOCC. Then, we show an infinite separation between SEP and PPT operations by providing a pair of states constructed from an unextendible product basis (UPB): they can be distinguished perfectly by PPT measurements, while the optimal error probability using SEP measurements admits an exponential lower bound. On the technical side, we prove this result by providing a quantitative version of the well-known statement that the tensor product of UPBs is UPB., Comments are welcome

In this work, we prove a novel one-shot ‘multi-sender’ decoupling theorem generalising Dupuis' seminal single sender decoupling theorem. We start off with a multipartite quantum state, say on $A_{1}A_{2}R$ , where $A_{1}, A_{2}$ are treated as the two ‘sender’ systems and $R$ is the reference system. We apply independent Haar random unitaries in tensor product on $A_{1}$ and $A_{2}$ and then send the resulting systems through a quantum channel. We want the channel output $B$ to be almost in tensor with the untouched reference $R$ . Our main result shows that this is indeed the case if suitable entropic conditions are met. An immediate application of our main result is to obtain a one-shot simultaneous decoder for sending quantum information over a $k$ -sender entanglement unassisted quantum multiple access channel (QMAC). The rate region achieved by this decoder is the natural one-shot quantum analogue of the pentagonal classical rate region. Assuming a simultaneous smoothing conjecture, this one-shot rate region approaches the optimal rate region of Yard et al. [20] in the asymptotic iid limit. Our work is the first one to obtain a non-trivial simultaneous decoder for the QMAC with limited entanglement assistance in both one-shot and asymptotic iid settings; previous works used unlimited entanglement assistance.

The tensor product of props was defined by Hackney and Robertson as an extension of the Boardman-Vogt product of operads to more general monoidal theories. Theories that factor as tensor products include the theory of commutative monoids and the theory of bialgebras. We give a topological interpretation (and vast generalisation) of this construction as a low-dimensional projection of a "smash product of pointed directed spaces". Here directed spaces are embodied by combinatorial structures called diagrammatic sets, while Gray products replace cartesian products. The correspondence is mediated by a web of adjunctions relating diagrammatic sets, pros, probs, props, and Gray-categories. The smash product applies to presentations of higher-dimensional theories and systematically produces higher-dimensional coherence cells., Comment: 74 pages

This paper presents the design and validation of a Tensor Product (TP)-based model of a family of nonlinear servo systems using an appropriate technique. Two parameters of the first principles state-space model of the servo system are optimally tuned using a metaheuristic Grey Wolf Optimizer algorithm in terms of several runs that lead to the parameter intervals. The derivation of the TP model starts with the linear parameter varying model of the servo system, which is next transformed to the strictly speaking TP model, inserted in a series connection with the servo system nonlinearity. The behaviors of the servo system, the TP model and the first principles model are tested in a different scenario to the parameter identification one, and the outputs are measured. The experimental results on a servo system laboratory equipment show that the TP model derived for this system ensures good performance in terms of small relative modeling errors.

Using the semi-tensor product method, this paper investigates how to represent the polynomial nonlinear system as the matrix and studies the stability of a class of the polynomial nonlinear systems. First, we represent polynomial nonlinear systems as the matrices via semi-tensor product method. Second, we construct and calculate Lyapunov function by using the semi-tensor product method. Third, we get a theorem of judging whether a class of polynomial nonlinear systems are asymptotically stable through semi-tensor product method. Finally, the study of an illustrative example shows that the new theorem of judging the stability of nonlinear systems in this paper is very effective.

This paper presents an approach for creating tensor product (TP) models based on sparse or non-uniform samples representing arbitrary datasets, and for manipulating the resulting TP structure to further identify the models behind the datasets. Given the usefulness of TP models in merging together closed algebraic formulae and tensor representations, it is argued that the proposed approach can be applied towards understanding the underlying complexity of a given dataset, while iteratively arriving at a model for the process that generated it. The key idea behind the approach is demonstrated using a dataset on sound pressure levels generated by different-sized airfoils in a wind tunnel.

This paper describes a Tensor Product (TP) based model via the inverted pendulum. Base of the controller designing is used the Linear Matrix Inequality (LMI) in order to stabilize the nonlinear system. This case shows the quasi-Linear Parameter-Varying (qLPV) modeling and the Higher-Order Singular Value Decomposition (HOSVD) based TP transformation.

The paper presents the tensor product based model of the highly nonlinear induction machine model. The independence of flux and speed state feedback control has been examined with integral action. Robustness of the designed controller has been investigated with applied load torque, additional measurement noise and parameter uncertainties.

While there has been a surge of recent interest in learning differential equation models from time series, methods in this area typically cannot cope with highly noisy data. We break this problem into two parts: (i) approximating the unknown vector field (or right-hand side) of the differential equation, and (ii) dealing with noise. To deal with (i), we describe a neural network architecture consisting of tensor products of one-dimensional neural shape functions. For (ii), we propose an alternating minimization scheme that switches between vector field training and filtering steps, together with multiple trajectories of training data. We find that the neural shape function architecture retains the approximation properties of dense neural networks, enables effective computation of vector field error, and allows for graphical interpretability, all for data/systems in any finite dimension $d$. We also study the combination of either our neural shape function method or existing differential equation learning methods with alternating minimization and multiple trajectories. We find that retrofitting any learning method in this way boosts the method's robustness to noise. While in their raw form the methods struggle with 1% Gaussian noise, after retrofitting, they learn accurate vector fields from data with 10% Gaussian noise., Comment: Accepted for publication in the Proceedings of the 54th Asilomar Conference on Signals, Systems, and Computers

We provide a deterministic $\tilde{O}(\log N)$-space algorithm for estimating random walk probabilities on undirected graphs, and more generally Eulerian directed graphs, to within inverse polynomial additive error ($\epsilon=1/\mathrm{poly}(N)$) where $N$ is the length of the input. Previously, this problem was known to be solvable by a randomized algorithm using space $O(\log N)$ (following Aleliunas et al., FOCS 79) and by a deterministic algorithm using space $O(\log^{3/2} N)$ (Saks and Zhou, FOCS 95 and JCSS 99), both of which held for arbitrary directed graphs but had not been improved even for undirected graphs. We also give improvements on the space complexity of both of these previous algorithms for non-Eulerian directed graphs when the error is negligible ($\epsilon=1/N^{\omega(1)}$), generalizing what Hoza and Zuckerman (FOCS 18) recently showed for the special case of distinguishing whether a random walk probability is $0$ or greater than $\epsilon$. We achieve these results by giving new reductions between powering Eulerian random-walk matrices and inverting Eulerian Laplacian matrices, providing a new notion of spectral approximation for Eulerian graphs that is preserved under powering, and giving the first deterministic $\tilde{O}(\log N)$-space algorithm for inverting Eulerian Laplacian matrices. The latter algorithm builds on the work of Murtagh et al. (FOCS 17) that gave a deterministic $\tilde{O}(\log N)$-space algorithm for inverting undirected Laplacian matrices, and the work of Cohen et al. (FOCS 19) that gave a randomized $\tilde{O}(N)$-time algorithm for inverting Eulerian Laplacian matrices. A running theme throughout these contributions is an analysis of "cycle-lifted graphs", where we take a graph and "lift" it to a new graph whose adjacency matrix is the tensor product of the original adjacency matrix and a directed cycle (or variants of one)., Comment: 52 pages. Fixed a bug in Lemma 5.8 that affected Theorem 5.9

This paper addresses the detection of noisy structures in the context of infrared microscopy using labeled undirected graph matching. The selection of robust features as labels is determinant in this case of study, where hard conditions are dealt with: few relevant topological information, a potentially huge number of outliers and low contrasted images resulting in noisy graphs. Firstly, the image texture is reliably caught through a scale, rotation and intensity invariant histogram of oriented gradients. Secondly, in structures presenting numerous symmetries, the graph shape is locally registered while discriminated using weight from a super-increasing series. Coupled to the flexibility of the graph tensor product-based similarity metric, the matching framework achieves satisfying results.

The inevitable uncertainties lying in spacecraft relative motion, such as navigation, modeling and mechanism execution, may deviate the actual trajectory from the nominal value and even pose a threat to mission security. This paper proposed an uncertainty propagation method based on generalized polynomial chaos expansion to quantify the impact of uncertainty on trajectory performance, and the collocation points in multi-dimensional space are generated by tensor product method. The proposed method and the conventional Monte Carlo simulation are compared and analyzed numerically. The result shows that the proposed method not only guarantees accuracy, but also implies higher computation efficiency.

The degeneracy problem of nonlinear feedback shift registers (NFSRs) is investigated in this paper. A new method for finding sub-families included in a family of NFSR sequences is proposed. First, the algebraic form of Fibonacci NFSR is proposed by viewing the NFSR as a Boolean network. Some properties of the algebraic expression are provided. Then, the observability of NFSRs is defined and the observability matrix is constructed, based on some explicit forms two algorithms for obtaining the observability matrix are obtained. A way to find sub-families of the given NFSR is derived and some sufficient and necessary conditions are obtained. Finally, an example is presented to depict the theoretical results.

In this paper, the L(p, q)-coloring problem of the graph is studied by using the semi-tensor product. First, by introducing two new logical operators a necessary and sufficient condition for solving the L(p, q)-coloring problem is given. Moreover, it is noted that all solutions of the obtained logical equations are corresponding to each coloring scheme. Second, the necessary and sufficient condition is converted to an algebraic form. Based on this, all coloring schemes can be obtained through searching all column index of the zero columns. Finally, an illustration example is given to show the effectiveness of the obtained results in this paper.

We propose a Lyapunov-based control method for stochastic quantum systems. The method can make the quantum state converge to a separable mixed target state from an arbitrary initial state. Specifically, the target state can be written as the tensor product of two matrices. Based on a well designed virtual mechanical quantity, a Lyapunov function and two control laws are presented: one can make the system equilibria in the largest invariant set stable according to LaSalle's invariance principle, and the other one can make the target state globally asymptotically stable.

One of the main challenges in building hybrid systems that integrate symbolic and sub-symbolic levels of computations is elaboration of methods that would allow expressing symbolic manipulations in a form of neural networks dynamics. In particular, it is extremely important to be able to process symbolic structures in a distributed format and perform analysis over this tensor representation that would be equivalent to a sequence of symbolic operations over the original structure. In this paper, the novel method is proposed for processing and analysis of semantics tree in a form of generated neural network. This method is built upon the Tensor Product Representations (TPRs) framework, scales well for an arbitrary structure and allows expressing simple symbolic operations in a form of neural network dynamics.

In terms of the tensor product of real Hilbert spaces, the paper provides the description of the results of a functional-geometric study of the necessary and sufficient conditions for the existence of a differential realization of a continuous infinite-dimensional dynamical system in the class of controlled bilinear nonstationary ordinary differential equations of the second order (including quasi-linear hyperbolic models) in a separable Hilbert space. Concurrently, the topological and metric conditions for the continuity of the Rayleigh – Ritz operator with the calculation of the fundamental group of its image are analytically substantiated. The results obtained give incentives for generalizations in the qualitative theory of nonlinear structural identification of higher order multi-linear differential models.

Tensorial two-dimensional principal component analysis (T2DPCA), a novel approach considering the third-order tensors as linear operators on the space of oriented matrices, has been proposed recently and showed better performance than traditional principal component analysis (PCA) in image analysis and recognition. Kernel T2DPCA (KT2DPCA), a nonlinear extension of T2DPCA, is a powerful technique for extracting nonlinear structure from data. However, the original form of KT2DPCA requires storing and manipulating the kernel matrix tensor the size of which is the square of the number of samples. This becomes computationally expensive when the number of samples is large. A similar problem occurs with T2DPCA. In this manuscript, we propose an incremental method to recursively estimate the multiple principal tensors. First, Oja’s rule is extended by replacing the general vector product with the defined tensor product, called T-product, in the tensor spaces. Then, to provide a sparse representation, we consider a tensor-based distance criterion. By combining the above two steps, incremental T2DPCA (IT2DPCA) is formulated. Finally, IT2DPCA is further extended to a nonlinear version by using the general kernel trick in machine learning field, but with a new inner product called inside product defined with the T-product in the tensor spaces, and simultaneously with the general inner product defended in vector spaces. Feasibility of the addressed methods is illustrated by experimental results on face recognition.

State-space modelling theory is still a current topic of electrical machines in drive control. Induction machines can be described by nonlinear model leading to linear parameter varying (LPV) state-space model. In order to solve the above-mentioned highly nonlinear problem, Tensor Product (TP) transformation has been chosen. Big advantage of TP modelling are the short simulation time in MATLAB and speed controller can be designed easily.

This paper presents an extension to the subspace-based direction-of-arrival (DOA) estimation algorithm dimensionality reduction multiple signal classification (DR-MUSIC). The extension operates with a single six-component vector sensor collecting multiple temporally displaced data sets. The temporal displacements invoke phase-shifts specific to the carrier frequencies of each signal to be located and are arranged into a matrix that is combined via tensor product with the arrival angle function steering matrix used in DR-MUSIC's iterative search. This procedure creates a more robust steering matrix that can resolve the directions of a number of signals greatly exceeding the number of antenna elements while rejecting interfering signals with impressive accuracy. Simulations demonstrate that the method also performs well under sub-zero SNR and SIR conditions.

Tensor product based model representation is a numerical technique to approximate the behavior of nonlinear systems. Based on the tensor product transformation a linear matrix inequality based controller design scheme can be applied efficiently. This framework is presented in the paper through the model of gantry crane.

This paper suggests two combinations of Tensor Product (TP)–based model transformation plus sliding mode control applied to the position control of nonlinear electromagnetic actuated clutch systems. Two cascade control system structures that consist of a TP–based controller in the inner control loop and a sliding mode–based controller in the outer control loop are presented. The proposed control system structures were tested on the nonlinear process model and validated by simulation results. A comparative analysis is given.

In this paper, we proposed a Semantic Tensor Product (STP) model aimed at finding a more grammatically correct caption for a given image. We tried two kinds of ways to integrate semantic concepts (i.e. tags) into Tensor Product Representation (TPR) in the process of image caption generation, where Tensor Product Representation has the advantage of decoding a role embedding using a corresponding unbinding vector in any position directly, while semantic concepts could provide crucial clues for internal TPR prediction. We qualitatively analyze the two proposed STP based models, and quantitatively evaluate them on benchmark dataset: MSCOCO. Experiment result shows that our results outperform the other models, and generated captions are more natural.

The paper presents a versatile library of analytic and quasi-analytic complex-valued wavelet packets (WPs) which originate from discrete splines of arbitrary orders. The real parts of quasi-analytic WPs are the regular spline-based orthonormal WPs designed in [1]. The imaginary parts are the so-called complementary orthonormal WPs, which, unlike the symmetric regular WPs, are antisymmetric. Tensor products of 1D quasi-analytic WPs provide a diversity of 2D WPs oriented in multiple directions. For example, the set of fourth-level WPs comprise 62 different directions. The designed computational scheme enables extremely fast and easy implementation of the WP transforms.

In this paper we use the STP theory (semi-tensor product of matrices) to consider the construction of the incompatible graph of finite state machines (FSMs) in a mathematical manner. First, the output dynamics of FSMs are formulated as a bilinear dynamic equation by expressing output symbols as vectors and based on the state transition equation developed by the authors recently. Second, with the output dynamic equation, we design an algebraic algorithm for constructing the incompatible graph of an FSM, where some algebraic results proposed by the authors are used, such as, an algebraic criterion and an algorithm of determining whether a pair of states is k-difference in the sense of language recognition. Examples are given to verify these results.

The concept of frames is used extensively for the representation of signal or functions. Recently, this concept is applied more and more for the representation of operators, both in theory as well as in the application for the numerical solutions of operator equations. In this paper we first give a survey about the matrix representation of operators using frames. Then we prove that the tensor product of frames forms a Banach frame and an atomic decomposition for the space of bounded operators of Hilbert spaces.

Low-rank matrix models have been universally useful for numerous applications starting from classical system identification to more modern matrix completion in signal processing and statistics. The Schatten-1 norm, also known as the nuclear norm, has been used as a convex surrogate of the low-rankness since it induces a low-rank solution to inverse problems. While the Schatten-1 norm for low-rankness has a nice analogy with the l 1 norm for sparsity through the singular value decomposition, other matrix norms also induce low-rankness. Particularly as one interprets a matrix as a linear operator between Banach spaces, various tensor product norms generalize the role of the Schatten-1 norm. Inspired by a recent work on the max-norm-based matrix completion, we provide a unified view on a class of tensor product norms and their interlacing relations on low-rank operators. Furthermore we derive entropy estimates between the injective and projective tensor products of a family of Banach space pairs and demonstrate their applications to matrix completion and decentralized subspace sketching.

Construction of directional compactly supported tensor product complex tight framelets cptTP-ℂTF 6 is discussed. The construction algorithm employs optimization techniques and puts extensive emphasis on frequency response and spatial localization of the underlying one-dimensional tight framelet filter banks. A concrete example of cptTP-ℂTF 6 is provided. Numerical experiments show that such constructed cptTP-ℂTF 6 have good performance in image denoising.

Using the theories of many-valued logic and semi-tensor product of matrices (STP), this paper investigates how to mathematically determine whether or not a regular language is recognized by a finite automaton. To this end, the behavior of finite automata is first formulated as bilinear dynamic equations, which provide a uniform model for deterministic and non-deterministic finite automata. Based on the bilinear model, the recognition capacity of finite automata understanding of regular languages is investigated and serval algebraic criteria are obtained. With the algebraic criteria, to judge whether a regular sentence is accepted by a finite automaton or not, one only need to calculate an STP of some vectors, rather than making the sentence run over the machine as traditional manners. Further, the inverse problem of recognition is considered, an algorithm is developed that can mathematically construct all the accepted sentences for a given finite automaton. The algebraic approach of this paper may be a new angle and means to understand and analyze the dynamics of finite automata.

A multidimensional linear algebra based image classification technique is proposed in this paper. The image on a particular class can be represented as a t-linear combination of images belong to same class whereas it is difficult to represent it using images in other classes. Motivated by this concept, the proposed method constructs tensors which provide approximate representation of each class during the training process. Lateral slices of these tensors are the images belongs to each class. In the next step, the strength of the representation coefficient is measured to determine the class where the image belongs. To achieve this, the proposed method computes a projection operator on to the set of subspaces throughout the learning process. Conventional method uses vectorization of images to form a data matrix. In contrast, the proposed method represents each class as a tensor for the effective utilization of structural information contained in the image samples. Feature extraction and classification is done in a single step. Moreover when compared to the predecessor, the proposed method requires a single projection operator which reduces the computational time to a large extent. The experimental results demonstrate the superiority of the proposed method over the existing method.

This paper proposes a semi-tensor product method for the controllability of multi-agent systems (MASs) over finite fields. The main tool is the algebraic state space representation (ASSR) approach. Firstly, the dynamics of MASs over finite fields is converted into an equivalent algebraic form by using the semi-tensor product of matrices. Secondly, a controllability matrix is constructed by using the algebraic form, based on which, a necessary and sufficient condition is presented for the controllability of MASs over finite fields. Thirdly, an algorithm is given to find the minimal number of leaders which makes MASs over finite fields controllable. Finally, an example is given to show the effectiveness of the obtained new results.

This paper presents the design and validation of two cascade control system (CCS) structures for the level control of vertical three tank systems. The first CCS structure consists of a state feedback controller in the inner control loop and a Proportional-Integral-Derivative (PID) controller in the outer control loop and the second CCS structure consists of a Tensor Product-based controller in the inner control loop and a PID controller in the outer control loop. The simulation results validate the suggested control structures Finally a comparative study is also given.

The derivation of a Tensor Product (TP)–based model of magnetic levitation systems using the TP–based model transformation method is proposed. The TP model approximates the behavior of the plant, but exhibiting a numerical approximation error. Finally, the behavior of the TP model is compared to the laboratory equipment behavior taking into consideration two testing scenarios. The experimental results obtained for the laboratory equipment show that the mean square error is smaller in the second testing scenario.