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The genetic code is a set of regulatory principles that control the translation of information encoded in messenger RNA (mRNA) into a sequence of amino acids. This study proposes a model starting from the relationships between its physicochemical properties of nucleobase in the codons. We employed a binary metric and the Kronecker product to represent the physicochemical properties of nucleobases in a vector space. This state space has a hierarchical order, with self-similarity and symmetry properties in the hydrogen bonds of triplets. The state space can be mapped under linear transformations to a toroidal geometry with allometric properties. Furthermore, this geometric representation highlights a charge symmetry that exists in amino acids and in the distribution of essential and non-essential amino acids. These results describe a state space between codon-anticodon interactions that can be interpreted as Bell states. This suggests that there is a quantum phenomenon involved in the mechanisms of information storage in DNA. In addition, toroidal geometry can be used to represent the sequences of codons of the mRNA that encode the sequence of amino acids of the proteins to find similarities or homologies between evolutionary related species. [ABSTRACT FROM AUTHOR]

A (signed) graph is said to exhibit the strong reciprocal (anti-reciprocal) eigenvalue property (SR) (resp., (-SR)) if for any eigenvalue λ\lambda , it has 1λ\frac{1}{\lambda } (resp.,−1λ-\frac{1}{\lambda }) as an eigenvalue as well, with the same multiplicity. It is well known that the corona of a (signed) graph does have the property -SR, and if the graph has symmetric spectrum, then it also has the property SR. Therefore, it is interesting to identify (signed) graphs which are not corona graphs with the properties SR or -SR. Recently, a few constructions for unsigned graphs with the property -SR have been offered. In this article, we extend such constructions to signed graphs.

We have considered a generalized Sylvester-transpose matrix equation $ AXB + CX^TD = E, $ where $ A, B, C, D, $ and $ E $ are given rectangular matrices over a generalized quaternion skew-field, and $ X $ is an unknown matrix. We have applied certain vectorizations and real representations to transform the matrix equation into a matrix equation over the real numbers. Thus, we have investigated a solvability condition, general exact/least-squares solutions, minimal-norm solutions, and the exact/least-squares solution closest to a given matrix. The main equation included the equation $ AXB = E $ and the Sylvester-transpose equation. Our results also covered such matrix equations over the quaternions, and quaternionic linear systems.

We consider local polynomial estimation for varying coefficient models and derive corresponding equivalent kernels that provide insights into the role of smoothing on the data and fill a gap in the literature. We show that the asymptotic equivalent kernels have an explicit decomposition with three parts: the inverse of the conditional moment matrix of covariates given the smoothing variable, the covariate vector, and the equivalent kernels of univariable local polynomials. We discuss finite-sample reproducing property which leads to zero bias in linear models with interactions between covariates and polynomials of the smoothing variable. By expressing the model in a centered form, equivalent kernels of estimating the intercept function are asymptotically identical to those of univariable local polynomials and estimators of slope functions are local analogues of slope estimators in linear models with weights assigned by equivalent kernels. Two examples are given to illustrate the weighting schemes and reproducing property. [ABSTRACT FROM AUTHOR]

Transformer has emerged as one of the modern neural networks that has been applied in numerous applications. However, transformers' large and deep architecture makes them computationally and memory-intensive. In this paper, we propose the block g-circulant matrices to replace the dense weight matrices in the feedforward layers of the transformer and leverage the DCT-DST algorithm to multiply these matrices with the input vector. Our test using Portuguese-English datasets shows that the suggested method improves model memory efficiency compared to the dense transformer but at the cost of a slight drop in accuracy. We found that the model Dense-block 1-circulant DCT-DST of 128 dimensions achieved the highest model memory efficiency at 22.14%. We further show that the same model achieved a BLEU score of 26.47%. [ABSTRACT FROM AUTHOR]

We develop a novel framework for the analysis of medical imaging data, including magnetic resonance imaging, functional magnetic resonance imaging, computed tomography and more. Medical imaging data differ from general images in two main aspects: (i) the sample size is often considerably smaller and (ii) the interpretation of the model is usually more crucial than predicting the outcome. As a result, standard methods such as convolutional neural networks cannot be directly applied to medical imaging analysis. Therefore, we propose the deep Kronecker network , which can adapt to the low sample size constraint and offer the desired model interpretation. Our approach is versatile, as it works for both matrix- and tensor-represented image data and can be applied to discrete and continuous outcomes. The deep Kronecker network is built upon a Kronecker product structure, which implicitly enforces a piecewise smooth property on coefficients. Moreover, our approach resembles a fully convolutional network as the Kronecker structure can be expressed in a convolutional form. Interestingly, our approach also has strong connections to the tensor regression framework proposed by Zhou et al. (2013) , which imposes a canonical low-rank structure on tensor coefficients. We conduct both classification and regression analyses using real magnetic resonance imaging data from the Alzheimer's Disease Neuroimaging Initiative to demonstrate the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

The operations in spectral fuzzy graphs such as Cartesian product, strong product, bipartite double, Kronecker product and Lexicographic product (composition) are analysed to elucidate the impact of eigenvalue spectra. Also, the properties are derived which delves the effect on spectral fuzzy graphs. [ABSTRACT FROM AUTHOR]

Copyright of Systems Engineering & Electronics is the property of Journal of Systems Engineering & Electronics Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We have considered a generalized Sylvester-transpose matrix equation A X B + C X T D = E , where A , B , C , D , and E are given rectangular matrices over a generalized quaternion skew-field, and X is an unknown matrix. We have applied certain vectorizations and real representations to transform the matrix equation into a matrix equation over the real numbers. Thus, we have investigated a solvability condition, general exact/least-squares solutions, minimal-norm solutions, and the exact/least-squares solution closest to a given matrix. The main equation included the equation A X B = E and the Sylvester-transpose equation. Our results also covered such matrix equations over the quaternions, and quaternionic linear systems. [ABSTRACT FROM AUTHOR]

A chain graph is a bipartite graph in which the neighborhood of the vertices in each partite set forms a chain with respect to set inclusion. By extending the concept of nesting from a bipartite graph to a k partite graph, a k-nested graph is defined. A half graph is a chain graph having no pairs of duplicate vertices. Similarly, a 'k-half graph' is a class of knested graph with no pairs of duplicate vertices. The (first) Zagreb matrix or Z-matrix denoted by Z(G) = (zij)n×n of a graph G, whose vertex vi has degree di is defined by zij = di + dj if the vertices vi and vj are adjacent and zij = 0 otherwise. Let ζ1, ζ2, . . ., ζn be the Zagreb eigenvalues of Z(G) and the Zagreb energy is the sum of the absolute values of the Zagreb eigenvalues. We obtain the determinant, eigenvalues and inverse of a k-half graph with respect to the Z-matrix. Bounds for the Zagreb energy and spectral radius are discussed along with the main and non-main Zagreb eigenvalues of a k-half graph. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we investigate the distinguishing chromatic number of Kronecker product of paths, cycles, star graphs, symmetric trees, almost symmetric trees, and bisymmetric trees.

In linear system identification problems, the Wiener filter represents a popular tool and stands as an important benchmark. Nevertheless, it faces significant challenges when identifying long-length impulse responses. In order to address the related shortcomings, the solution presented in this paper is based on a third-order tensor decomposition technique, while the resulting sets of Wiener–Hopf equations are solved with the conjugate gradient (CG) method. Due to the decomposition-based approach, the number of coefficients (i.e., the parameter space of the filter) is greatly reduced, which results in operating with smaller data structures within the algorithm. As a result, improved robustness and accuracy can be achieved, especially in harsh scenarios (e.g., limited/incomplete sets of data and/or noisy conditions). Besides, the CG-based solution avoids matrix inversion operations, together with the related numerical and complexity issues. The simulation results are obtained in a network echo cancellation scenario and support the performance gain. In this context, the proposed iterative Wiener filter outperforms the conventional benchmark and also some previously developed counterparts that use matrix inversion or second-order tensor decompositions. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Zhejiang University (Science Edition) is the property of Journal of Zhejiang University (Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we investigate when the generalized join product, generalized corona product, generalized edge corona product and the Kronecker product of graphs are divisor graphs. Also the situations under which the zero divisor graph and the co-maximal graph are divisor graphs are characterized. [ABSTRACT FROM AUTHOR]

This paper introduces a pioneering model that merges coupled Markov chains (CMC) with the fuzzy analytic hierarchy process (FAHP) to enhance multi-criteria decision making (MCDM) amidst the dynamic interplay of criteria. Traditional MCDM frameworks often lack the granularity to manage the intricate and changing relationships among criteria. Our model addresses this gap by integrating fuzzy numbers into AHP, providing a nuanced means to handle the inherent uncertainty of decision criteria. The application of the Kronecker product in CMC enriches our approach, offering a data-driven analysis while mitigating excessive dependence on subjective expert opinion. A comprehensive numerical example underlines the model's improved decision-making accuracy and efficiency, marking a substantial advancement in MCDM methodologies. This research contributes to the field by presenting a sophisticated yet practical framework for dynamic decision analysis in complex uncertain environments. [ABSTRACT FROM AUTHOR]

This paper presents a bilinear Chebyshev pseudo-spectral method to compute European and American option prices under the two-asset Black–Scholes and Heston models. We expand a function and its derivatives into their Chebyshev series, so the differentiation matrices that act on the Chebyshev coefficients are sparse and better conditioned. First, the equation is spatially discretized using a bilinear pseudo-spectral method created by Chebyshev polynomials and a first–order matrix differential equation (MDE) is obtained. Then, using the Kronecker product, this equation is converted to a system of first–order ODEs. For solving the European options, by using an eigenvalue decomposition method, the arising system will be analytically solved. Therefore, the arising errors are because of spatial discretization and quadrature errors. For solving the American options, the approach is combined with the operator splitting method or penalty method to obtain the temporal discretization. By avoiding some transforms to convert the equation to a constant coefficient equation without mixed derivatives, we will obtain more accurate solutions. Also, transforming the European option into a set of separated ODEs by an eigenvalue decomposition causes to reduce the computational complexity. We also consider the hedge ratios which show the sensitivity of an option to the stock prices. Several numerical examples are included to show the accuracy and efficiency of the proposed approach. The results show that the spectral convergence can be achieved for models with smooth functions. [ABSTRACT FROM AUTHOR]

A visually meaningful double-image image encryption algorithm is proposed based on a fractional-order chaotic system combined with 2D compressive sensing (CS). Specifically, in the pre-encryption phase, a 2D discrete wavelet transform (DWT) is performed on two grayscale images followed by a measurement using 2D CS. The measurement matrices are generated by using the fractional-chaotic system and Kronecker product (KP). Then, a Zigzag confusion operation is performed on the two generated measurement matrices after quantization to obtain two secret images. A new measurement matrix optimization algorithm is presented to optimize the high-dimensional measurement matrix. In the embedding phase, a new repairable embedding algorithm is designed which combines matrix encoding and 2K correction algorithm. The loss of the carrier image quality is effectively reduced while the robustness of the cipher image is enhanced. It is more suitable for practical applications. Numerical simulation and performance analyses show that the average PSNR value of visually meaningful cipher images is 41.7 dB, and the PSNR value of the reconstructed image is 1 ~ 3 dB higher than those in other studies. In addition, the proposed algorithm is also superior to some recently proposed algorithms in efficiency, security, and robustness. [ABSTRACT FROM AUTHOR]

The genetic code is a set of regulatory principles that control the translation of information encoded in messenger RNA (mRNA) into a sequence of amino acids. This study proposes a model starting from the relationships between its physicochemical properties of nucleobase in the codons. We employed a binary metric and the Kronecker product to represent the physicochemical properties of nucleobases in a vector space. This state space has a hierarchical order, with self-similarity and symmetry properties in the hydrogen bonds of triplets. The state space can be mapped under linear transformations to a toroidal geometry with allometric properties. Furthermore, this geometric representation highlights a charge symmetry that exists in amino acids and in the distribution of essential and non-essential amino acids. These results describe a state space between codon-anticodon interactions that can be interpreted as Bell states. This suggests that there is a quantum phenomenon involved in the mechanisms of information storage in DNA. In addition, toroidal geometry can be used to represent the sequences of codons of the mRNA that encode the sequence of amino acids of the proteins to find similarities or homologies between evolutionary related species.

The Kronecker product of two matrices is known as a special algebraic operation of two arbitrary matrices in the computational aspect of matrix theory. This kind of matrix operation has some interesting and striking operation properties, one of which is given by $ (A \otimes B)(C \otimes D) = (AC) \otimes (BD) $ and is often called the mixed-product equality. In view of this equality, the Kronecker product $ A_1 \otimes A_2 $ of any two matrices can be rewritten as the dilation factorization $ A_1 \otimes A_2 = (A_1 \otimes I_{m_2})(I_{n_1} \otimes A_2) $, and the Kronecker product $ A_1 \otimes A_2 \otimes A_3 $ can be rewritten as the dilation factorization $ A_1 \otimes A_2 \otimes A_3 = (A_1\otimes I_{m_2} \otimes I_{m_3})(I_{n_1} \otimes A_2 \otimes I_{m_3})(I_{n_1} \otimes I_{n_2} \otimes A_3) $. In this article, we proposed a series of concrete problems regarding the dilation factorizations of the Kronecker products of two or three matrices, and established a collection of novel and pleasing equalities, inequalities, and formulas for calculating the ranks, dimensions, orthogonal projectors, and ranges related to the dilation factorizations. We also present a diverse range of interesting results on the relationships among the Kronecker products $ I_{m_1} \otimes A_2 \otimes A_3 $, $ A_1 \otimes I_{m_2} \otimes A_3 $ and $ A_1 \otimes A_2 \otimes I_{m_3} $.

The left/right projection matrix and recovery matrix used for the reconstruction error in the traditional generalized low-rank approximation of matrix models are the same and orthogonal, which makes the model inflexible. To this end, we propose the flexible sparse robust low-rank approximation of matrices model to integrate feature selection into subspace learning and to exclude the redundant features. In the proposed model, two recovery matrices are introduced to together recover the original image data from the subspace spanned by the selected features, resulting in more freedom and flexible to jointly select useful features for low-dimensional representation. Moreover, the L1-norm is imposed on the reconstruction error and L2,1-norm on the Kronecker product of left and right transformation matrices, which can reduce the influence of noise on errors and perform feature selection while learning the optimal transformation matrices and recovery matrices. According to some theoretical analysis, an alternative iterative solution method is designed, and the convergence and time complexity of the algorithm are analyzed. The experimental results on some image datasets show that our method is superior to the existing state-of-the-art methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider the aeroelastic flutter problem (AEFP) in terms of matrix equations. We provide a general framework on the spectral theory of three parameter AEFP in tensor product space under the auspices of the techniques of multiparameter eigenvalue problems (MEP) in matrix form. It is to noted that, the AEFP of an undamped system can be converted to a linear singular two parameter eigenvalue problem (2PEP) in double dimension and by quasilinearization technique it can be converted to a linear three parameter eigenvalue problem (3PEP) of the same dimension. The de-facto way to find the numeric of MEP is by solving associated joint generalised eigenvalue problem (GEP) using conventional numerical method, provided the problem being nonsingular. Since the transformed version of AEFP in matrix form is singular, so the usual solution techniques for MEP can not be applied to address AEFP and it necessitates to adopt alternate numerical techniques for the same. In the current paper, it is intended to address singular AEFP under certain assumptions. The paper also serves as a report on the Kronecker product method, so called Delta method developed by Atkinson for finding finite eigenvalues of singular three parameter AEFP. [ABSTRACT FROM AUTHOR]

We introduce a simple algorithm that efficiently computes tensor products of Pauli matrices. This is done by tailoring the calculations to this specific case, which allows to avoid unnecessary calculations. The strength of this strategy is benchmarked against state-of-the-art techniques, showing a remarkable acceleration. As a side product, we provide an optimized method for one key calculus in quantum simulations: the Pauli basis decomposition of Hamiltonians. [ABSTRACT FROM AUTHOR]

In control and communications, the phase-locked loop (PLL) is regarded as the demodulator. Under the presence of small noise, the PLL system fails to accomplish the locking condition resulting in the stochastic phase difference. As a result of this, the PLL becomes a nontrivial nonlinear stochastic system. To circumvent the curse of dimensionality and nonlinearity, we exploit the method of linearization in the Carleman framework in combination with the finite closure for the stochastic system considered here. We show that the Carleman linearization has proven useful to preserve the nonlinearity via bilinearization. The Carleman setup of the nonlinear stochastic differential system has the Markov property and the terms are manageable. Then, we filter the states of the PLL using the filtering theory of the homogeneous Markov process. Finally, the numerical simulations reveal the superiority of the proposed filtering in Carleman setting in contrasts with the celebrated extended Kalman filtering framework. [ABSTRACT FROM AUTHOR]

Discrete state Markov chains are widely used in regional science and economics to model spatial dynamics of a host of processes. This paper introduces an approach that extends the classic univariate focus to consider the joint spatial dynamics of two variables based on a Kronecker product framework. A test for interactions between the processes is proposed. The paper illustrates the framework in an empirical case study involving race and class dynamics in San Diego, California, and in a number of Monte Carlo simulations. The results reveal insights about these intersectional dynamics beyond those obtainable with the application of the traditional univariate Markov chain approach. [ABSTRACT FROM AUTHOR]

This paper provides two direct methods for solving the split quaternion matrix equation (AXB,CXD)=(E,F), where X is an unknown split quaternion η‐Hermitian matrix, and A, B, C, D, E, F are known split quaternion matrices with suitable size. Our tools are the Kronecker product, Moore–Penrose generalized inverse, real representation, and complex representation of split quaternion matrices. Our main work is to find the necessary and sufficient conditions for the existence of a solution of the matrix equation mentioned above, derive the explicit solution representation, and provide four numerical algorithms and two numerical examples. [ABSTRACT FROM AUTHOR]

The decomposition or approximation of a linear operator on a matrix space as a sum of Kronecker products plays an important role in matrix equations and low-rank modeling. The approximation problem in Frobenius norm admits a well-known solution via the singular value decomposition. However, the approximation problem in spectral norm, which is more natural for linear operators, is much more challenging. In particular, the Frobenius norm solution can be far from optimal in spectral norm. We describe an alternating optimization method based on semidefinite programming to obtain high-quality approximations in spectral norm, and we present computational experiments to illustrate the advantages of our approach. [ABSTRACT FROM AUTHOR]

Let $ \mathbb{D}^{m \times n} = \{A = A_{1}+\varepsilon A_{2}|A_{1}, A_{2}\in \mathbb{R}^{m \times n}\} $ be the set of all $ m\times n $ real dual matrices. In this paper, the following problems are considered. Problem I: Given dual matrices $ A = A_{1}+\varepsilon A_{2}\in \mathbb{D}^{m\times n} $ and $ B = B_{1}+\varepsilon B_{2}\in \mathbb{D}^{n\times n} $, find $ X\in S $ such that the dual matrix equation $ A^\top XA = B $ is satisfied, where $ S = \{X\in \mathbb{D}^{m \times m}|CX = D, C, D\in \mathbb{D}^{p \times m}\} $. Problem II: Given dual matrices $ A = A_{1}+\varepsilon A_{2}\in \mathbb{D}^{m\times n}, B = B_{1}+\varepsilon B_{2}\in \mathbb{D}^{n\times n} $ and $ \tilde{X} = \tilde{X}_{1}+\varepsilon \tilde{X}_{2}\in \mathbb{D}^{m\times m} $, with $ B_{i} = B^\top_{i}, i = 1, 2 $, find $ \hat{X}\in T $ such that $ \|\hat{X}-\tilde{X}\|_{{\rm D}} = \mathop{\min}\limits_{X\in T} \|X-\tilde{X}\|_{{\rm D}} = \mathop{\min}\limits_{X\in T}\sqrt{\Vert X_{1}-\tilde{X}_{1} \Vert^{2}+\Vert X_{2}-\tilde{X}_{2}\Vert^{2}} $, where $ T = \{X = X_{1}+\varepsilon X_{2}\in \mathbb{D}^{m \times m}|A^\top XA = B \ \ \mbox{s. t.} \ X_{i} = X^\top_{i}, i = 1, 2\} $. We derive the solvability conditions and the representation of the general solution of Problem I using the Moore-Penrose inverse. Also, we deduce the solvability conditions and the explicit formula of $ T $ and the unique approximation solution $ \hat{X} $ of Problem II by applying the Moore-Penrose inverse and Kronecker product of matrices. Finally, we give a numerical example to show the correctness of our result.

Transformer has emerged as one of the modern neural networks that has been applied in numerous applications. However, transformers’ large and deep architecture makes them computationally and memory-intensive. In this paper, we propose the block g-circulant matrices to replace the dense weight matrices in the feedforward layers of the transformer and leverage the DCT-DST algorithm to multiply these matrices with the input vector. Our test using Portuguese-English datasets shows that the suggested method improves model memory efficiency compared to the dense transformer but at the cost of a slight drop in accuracy. We found that the model Dense-block 1-circulant DCT-DST of 128 dimensions achieved the highest model memory efficiency at 22.14%. We further show that the same model achieved a BLEU score of 26.47%.

In the encoding and decoding process of transformer neural networks, a weight matrix-vector multiplication occurs in each multihead attention and feed forward sublayer. Assigning the appropriate weight matrix and algorithm can improve transformer performance, especially for machine translation tasks. In this study, we investigate the use of the real block-circulant matrices and an alternative to the commonly used fast Fourier transform (FFT) algorithm, namely, the discrete cosine transform–discrete sine transform (DCT-DST) algorithm, to be implemented in a transformer. We explore three transformer models that combine the use of real block-circulant matrices with different algorithms. We start from generating two orthogonal matrices, U and Q. The matrix U is spanned by the combination of the reals and imaginary parts of eigenvectors of the real block-circulant matrix, whereas Q is defined such that the matrix multiplication QU can be represented in the shape of a DCT-DST matrix. The final step is defining the Schur form of the real block-circulant matrix. We find that the matrix-vector multiplication using the DCT-DST algorithm can be defined by assigning the Kronecker product between the DCT-DST matrix and an orthogonal matrix in the same order as the dimension of the circulant matrix that spanned the real block circulant. According to the experiment's findings, the dense-real block circulant DCT-DST model with largest matrix dimension was able to reduce the number of model parameters up to 41%. The same model of 128 matrix dimension gained 26.47 of BLEU score, higher compared to the other two models on the same matrix dimensions.

A constructive algorithm to compute elimination and duplication matrices for the operation of vectorization when is proposed. The matrix , obtained according to such algorithm, allows one to form a vector that contains only unique elements of the mentioned Kronecker product. In its turn, the matrix is for the inverse transformation. A software implementation of the procedure to compute the matrices and is developed. On the basis of the mentioned results, a new operation is defined for in case and its properties are studied. The difference and advantages of the developed operation in comparison with the known ones and () in case of vectorization of when are demonstrated. Using parameterization of the algebraic Riccati equation as an example, the efficiency of the operation to reduce overparameterization of the unknown parameter identification problem is shown. [ABSTRACT FROM AUTHOR]

This work focuses on linear system identification problems in the framework of the Wiener filter. Specifically, it addresses the challenging identification of systems characterized by impulse responses of long length, which poses significant difficulties due to the existence of large parameter space. The proposed solution targets a dimensionality reduction of the problem by involving the decomposition of a fourth-order tensor, using low-rank approximations in conjunction with the nearest Kronecker product. In addition, the rank of the tensor is controlled and limited to a known value without involving any approximation technique. The final estimate is obtained based on a combination of four (shorter) optimal filters, which are alternatively iterated. As a result, the designed iterative Wiener filter outperforms the traditional counterpart, being more robust to the accuracy of the statistics' estimates and/or noisy conditions. In addition, simulations performed in the context of acoustic echo cancellation indicate that the proposed iterative Wiener filter that exploits this fourth-order tensor decomposition achieves better performance as compared to some previously developed solutions based on lower decomposition levels. This study could further lead to the development of computationally efficient tensor-based adaptive filtering algorithms. [ABSTRACT FROM AUTHOR]

Let S R Ω n × n be the set of all n × n symmetric matrices on subspace Ω , where Ω = { z ∈ R n | G z = 0 , G ∈ R k × n }. The necessary and sufficient conditions for the matrix equations A X = B and X C = D to have a common solution in S R Ω n × n and also an expression for the general common solution are obtained. Further, the associated optimal approximate problem to a given matrix X ~ ∈ R n × n is discussed and the optimal approximate solution is elucidated. Finally, a numerical experiment is presented to validate the accuracy of our result. [ABSTRACT FROM AUTHOR]

Discovering underlying low dimensional structure of a high-dimensional matrix is traditionally done through low rank matrix approximations in the form of a sum of rank-one matrices. In this article, we propose a new approach. We assume a high-dimensional matrix can be approximated by a sum of a small number of Kronecker products of matrices with potentially different configurations, named as a hybrid Kronecker outer Product Approximation (hKoPA). It provides an extremely flexible way of dimension reduction compared to the low-rank matrix approximation. Challenges arise in estimating a hKoPA when the configurations of component Kronecker products are different or unknown. We propose an estimation procedure when the set of configurations are given, and a joint configuration determination and component estimation procedure when the configurations are unknown. Specifically, a least squares backfitting algorithm is used when the configurations are given. When the configurations are unknown, an iterative greedy algorithm is developed. Both simulation and real image examples show that the proposed algorithms have promising performances. Some identifiability conditions are also provided. The hybrid Kronecker product approximation may have potentially wider applications in low dimensional representation of high-dimensional data. for this article are available online. [ABSTRACT FROM AUTHOR]

This work presents a numerical algorithm based on the cubic B-spline quasi-interpolation method for the simulation of one and two-dimensional hyperbolic partial differential equations. In this method, cubic B-spline quasi-interpolation method is used for the approximation of spatial derivatives which produces a system of second-order ordinary differential equations. Further, the obtained system is decoupled into a system of first-order ordinary differential equations, and then forward difference approximation for time derivative is used to get final solutions. Some well-known problems from the literature are considered to check the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Orthogonal array is one of the important research subjects in combinatorial design theory and experimental design theory, and it is widely applied to statistics, computer science, coding theory and cryptography. There are many constructions and results for orthogonal array of strength 2, however the results on orthogonal array of strength t ≥ 3 are rare. In this paper, we first present two new effective constructions for orthogonal arrays of strength t ≥ 3 based on large sets of orthogonal arrays. Second, many infinite families of large sets of orthogonal arrays are obtained and then some new series of orthogonal array of strength t ≥ 3 are produced. [ABSTRACT FROM AUTHOR]

The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, we show that the quantum Fourier transform (QFT) can be derived by further decomposing the diagonal factors of the FFT matrix decomposition into products of matrices with Kronecker product structure. We analyze the implication of this Kronecker product structure on the discrete Fourier transform of rank-1 tensors on a classical computer. We also explain why such a structure can take advantage of an important quantum computer feature that enables the QFT algorithm to attain an exponential speedup on a quantum computer over the FFT algorithm on a classical computer. Further, the connection between the matrix decomposition of the DFT matrix and a quantum circuit is made. We also discuss a natural extension of a radix-2 QFT decomposition to a radix-d QFT decomposition. No prior knowledge of quantum computing is required to understand what is presented in this paper. Yet, we believe this paper may help readers to gain some rudimentary understanding of the nature of quantum computing from a matrix computation point of view.

The space-time structure of a teak wood specific gravity (SG) dataset was analyzed using a mixed-effects model. Spatial correlation increased in space, a phenomenon attributable to the maturation of apical meristems, while the temporal correlation of vascular meristems decreased over time. The decay of temporal correlation over time was attributed to the diminishing crown effect on the later formed wood further away from the pith, morphogen gradient, and probably changing microenvironmental conditions. The Kronecker product was used to collect spatiotemporal data on the intricate dynamic process of the evolution of the apical and lateral meristems. The results showed that height and relative radial distance (RRD) (i.e., the flow of time with wood formation) were statistically significant factors, with their interaction showing no significance. The results confirm the usefulness of using the space-time approach to elucidate the interaction between the apical and lateral meristems, two major inherent biological systems that control tree growth and wood formation dynamics. To understand the origins of patterns that vary both temporally and spatially in the tree, future work should describe the variation of SG within the tree due to increasing height (space) and diameter (age) as a matrix; then the correlation function can be modelled.

This document is a corrigendum for an article titled "Algebraic and toroidal representation of the genetic code" published in the journal Frontiers in Applied Mathematics & Statistics. The corrigendum acknowledges an error in the legend for Figure 8, which depicts toroidal mapping in polar coordinates showing the formation of a vortex in the direction of amino acids that use codons with strong or weak hydrogen bonds. The authors apologize for the error and state that it does not affect the scientific conclusions of the article. The document also includes a note from the publisher clarifying that the views expressed in the article are solely those of the authors and do not necessarily represent the views of their affiliated organizations or the publisher. The article was authored by Rodrigo Rodríguez-Gutierrez, Francisco Hernandez-Cabrera, Francisco Javier Almaguer-Martínez, and Jose De Jesus Bernal-Alvarado. Correspondence regarding the article can be directed to Francisco Hernandez-Cabrera. [Extracted from the article]

In linear system identification problems, the Wiener filter represents a popular tool and stands as an important benchmark. Nevertheless, it faces significant challenges when identifying long-length impulse responses. In order to address the related shortcomings, the solution presented in this paper is based on a third-order tensor decomposition technique, while the resulting sets of Wiener–Hopf equations are solved with the conjugate gradient (CG) method. Due to the decomposition-based approach, the number of coefficients (i.e., the parameter space of the filter) is greatly reduced, which results in operating with smaller data structures within the algorithm. As a result, improved robustness and accuracy can be achieved, especially in harsh scenarios (e.g., limited/incomplete sets of data and/or noisy conditions). Besides, the CG-based solution avoids matrix inversion operations, together with the related numerical and complexity issues. The simulation results are obtained in a network echo cancellation scenario and support the performance gain. In this context, the proposed iterative Wiener filter outperforms the conventional benchmark and also some previously developed counterparts that use matrix inversion or second-order tensor decompositions.

This paper introduces a pioneering model that merges coupled Markov chains (CMC) with the fuzzy analytic hierarchy process (FAHP) to enhance multi-criteria decision making (MCDM) amidst the dynamic interplay of criteria. Traditional MCDM frameworks often lack the granularity to manage the intricate and changing relationships among criteria. Our model addresses this gap by integrating fuzzy numbers into AHP, providing a nuanced means to handle the inherent uncertainty of decision criteria. The application of the Kronecker product in CMC enriches our approach, offering a data-driven analysis while mitigating excessive dependence on subjective expert opinion. A comprehensive numerical example underlines the model’s improved decision-making accuracy and efficiency, marking a substantial advancement in MCDM methodologies. This research contributes to the field by presenting a sophisticated yet practical framework for dynamic decision analysis in complex uncertain environments.

In this paper, we propose and analyze the numerical algorithms for fast solution of periodic elliptic problems in random media in ℝd$$ {\mathbb{R}}^d $$, d=2,3$$ d=2,3 $$. Both the two‐dimensional (2D) and three‐dimensional (3D) elliptic problems are considered for the jumping equation coefficients built as a checkerboard type configuration of bumps randomly distributed on a large L×L$$ L\times L $$, or L×L×L$$ L\times L\times L $$ lattice, respectively. The finite element method discretization procedure on a 3D n×n×n$$ n\times n\times n $$ uniform tensor grid is described in detail, and the Kronecker tensor product approach is proposed for fast generation of the stiffness matrix. We introduce tensor techniques for the construction of the low Kronecker rank spectrally equivalent preconditioner in a periodic setting to be used in the framework of the preconditioned conjugate gradient iteration. The discrete 3D periodic Laplacian pseudo‐inverse is first diagonalized in the Fourier basis, and then the diagonal matrix is reshaped into a fully populated third‐order tensor of size n×n×n$$ n\times n\times n $$. The latter is approximated by a low‐rank canonical tensor by using the multigrid Tucker‐to‐canonical tensor transform. As an example, we apply the presented solver in numerical analysis of stochastic homogenization method where the 3D elliptic equation should be solved many hundred times, and where for every random sampling of the equation coefficient one has to construct the new stiffness matrix and the right‐hand side. The computational characteristics of the presented solver in terms of a lattice parameter L$$ L $$ and the grid‐size, nd$$ {n}^d $$, in both 2D and 3D cases are illustrated in numerical tests. Our solver can be used in various applications where the elliptic problem should be solved for a number of different coefficients for example, in many‐particle dynamics, protein docking problems or stochastic modeling. [ABSTRACT FROM AUTHOR]

We propose and study structured time‐dependent inverse regression (STIR), a novel sufficient dimension reduction model, to analyze longitudinally measured, correlated biomarkers in relation to an outcome. The time structure is accommodated in an inverse regression model for the markers that can be applied both to equally and unequally spaced time points for each sample. The inverse regression structure also naturally accommodates retrospectively sampled markers, that is, markers measured in case‐control studies. We estimate the corresponding linear combinations of the markers, the reduction, using least squares. We show that under additional distributional assumptions the reduction contains sufficient information about the outcome. In extensive simulations the STIR linear combinations perform well in predictive models based on samples of realistic size. A Wald‐type test for association of a particular marker with outcome at any time point based on the STIR reduction has better power overall than assessing associations based on logistic or linear regression models that include all longitudinally measured markers as independent predictors. As illustrations we estimate the STIR reductions for a cohort study of diabetes and hyperlipidemia and a case‐control study of brain cancer with multiple longitudinally measured biomarkers. We assess the STIR reductions' predictive performance and identify outcome‐associated biomarkers. [ABSTRACT FROM AUTHOR]

In this paper, a direct solution method is proposed for solving generalized stochastic eigenvalue problems of structures with random uncertainties in structural parameters. Taking random parameters into consideration, predicting the statistical characteristics of the eigenvalues of structures within a probabilistic framework precisely is significant for structural analysis and design. However, the most typical method, the Monte‐Carlo Simulation is highly time consuming. And most of the existing methods with high computational efficiency compared to the Monte‐Carlo Simulation are approximate methods and may not be applicable for high‐precision requirements. Within the proposed method, the solution equations are deduced in terms of the properties of matrix operations step by step based on the Kronecker product and straightening operation, and thus the mean values and variances of the eigenvalues can be obtained directly with high precision. Three numerical examples are employed to illustrate the feasibility and applicability of the proposed method in specific engineering structures. The results indicate that the proposed method yields consistent mean values and variances of the eigenvalues with those obtained by the Monte‐Carlo Simulation respectively with significantly less computation time showing its superiority in accuracy and efficiency. [ABSTRACT FROM AUTHOR]

The space-time structure of a teak wood specific gravity (SG) dataset was analyzed using a mixed-effects model. Spatial correlation increased in space, a phenomenon attributable to the maturation of apical meristems, while the temporal correlation of vascular meristems decreased over time. The decay of temporal correlation over time was attributed to the diminishing crown effect on the later formed wood further away from the pith, morphogen gradient, and probably changing microenvironmental conditions. The Kronecker product was used to collect spatiotemporal data on the intricate dynamic process of the evolution of the apical and lateral meristems. The results showed that height and relative radial distance (RRD) (i.e., the flow of time with wood formation) were statistically significant factors, with their interaction showing no significance. The results confirm the usefulness of using the space-time approach to elucidate the interaction between the apical and lateral meristems, two major inherent biological systems that control tree growth and wood formation dynamics. To understand the origins of patterns that vary both temporally and spatially in the tree, future work should describe the variation of SG within the tree due to increasing height (space) and diameter (age) as a matrix; then the correlation function can be modelled. [ABSTRACT FROM AUTHOR]

The distance, distance signless Laplacian and distance Laplacian matrix of a simple connected graph G , are denoted by D (G) , D Q (G) = D (G) + Tr (G) and D L (G) = Tr (G) − D (G) , respectively, where Tr (G) is the diagonal matrix of vertex transmission. Ger s ̌ gorin discs for any n × n square matrix A = [ a i j ] are the discs { z ∈ ℂ : | z − a i i | ≤ R i (A) } , where R i (A) = ∑ j ≠ i | a i j | , i = 1 , 2 , ... , n. The famous Ger s ̌ gorin disc theorem says that all the eigenvalues of a square matrix lie in the union of the Ger s ̌ gorin discs of that matrix. In this paper, some classes of graphs are studied for which the smallest Ger s ̌ gorin disc contains every distance and distance signless Laplacian eigenvalues except the spectral radius of the corresponding matrix. For all connected graphs, a lower bound and for trees, an upper bound of every distance signless Laplacian eigenvalues except the spectral radius is given in this paper. These bounds are comparatively better than the existing bounds. By applying these bounds, we find some infinite classes of graphs for which the smallest Ger s ̌ gorin disc contains every distance signless Laplacian eigenvalues except the spectral radius of the distance signless Laplacian matrix. For the distance Laplacian eigenvalues, we have given an upper bound and then find a condition for which the smallest Ger s ̌ gorin disc contains every distance Laplacian eigenvalue of the distance Laplacian matrix. These results give partial answers from some questions that are raised in [2]. [ABSTRACT FROM AUTHOR]

This paper presents an efficient fast Walsh-Hadamard-Hartley transform (FWHT) algorithm that incorporates the computation of the Walsh-Hadamard transform (WHT) with the discrete Hartley transform (DHT) into an orthogonal, unitary single fast transform possesses the block diagonal structure. The proposed algorithm is implemented in an integrated butterfly structure utilizing the sparse matrices factorization approach and the Kronecker (tensor) product technique, which proved a valuable and fast tool for developing and analyzing the proposed algorithm. The proposed approach was distinguished by ease of implementation and reduced computational complexity compared to previous algorithms, which were based on the concatenation of WHT and FHT by saving up to 3N-4 of real multiplication and 7.5N-10 of real addition. [ABSTRACT FROM AUTHOR]