Conjugate Transpose Research Articles

A SIMPLIFIED FREQUENCY DOMAIN BEAMFORMING APPROACH FOR ACTIVE AND PASSIVE SURFACE WAVE SURVEYS

Frequency domain beamforming (FDBF) and cylindrical frequency domain beamforming (CFDBF) techniques are among the most robust methods for extracting experimental phase velocity dispersion data from wavefields recorded during either active- or passive-source surface wave testing. Both FDBF and CFDBF transformations rely on calculating the spatiospectral correlation matrix to compute the steered power response spectrum, which is essential for extracting surface wave dispersion data. The dimensions of the spatiospectral correlation matrix are directly influenced by the frequency sampling and the number of sensors employed in the survey setup. When the number of sensors is large, and/or the frequency range is broad, and/or the frequency sampling is dense, the spatiospectral correlation matrix size leads to a greater number of required computations to calculate the steered power response spectrum for both the FDBF and CFDBF techniques. Therefore, we propose a simplified and more computationally efficient frequency domain beamforming algorithm for both active and passive surface wave surveys. This method avoids the calculation of the spatiospectral correlation matrix and its multiplication with the Hermitian transpose of the steering vector, thereby reducing the computational complexity by an order. Furthermore, by leveraging vectorization techniques, the proposed algorithm employs matrix multiplications instead of loops, significantly heightening computational efficiency. We rigorously tested the algorithm using surface wave datasets collected across diverse subsurface conditions with varying sensor arrays and acquisition parameters (e.g., sampling frequencies and window lengths) to evaluate its efficacy. The results demonstrate that our proposed algorithm outperforms the conventional FDBF method in the processing time, highlighting its practical effectiveness. This method shows promise for integration into surface wave data processing software, offering substantial improvements in computational efficiency without compromising accuracy

Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations

Abstract In this article, we present two new algorithms referred to as the improved modified gradient-based iterative (IMGI) algorithm and its relaxed version (IMRGI) for solving the complex conjugate and transpose (CCT) Sylvester matrix equations, which often arise from control theory, system theory, and so forth. Compared with the gradient-based iterative (GI) (A.-G. Wu, L.-L. Lv, and G.-R. Duan, Iterative algorithms for solving a class of complex conjugate and transpose matrix equations, Appl. Math. Comput. 217 (2011), 8343–8353) and the relaxed GI (RGI) (W.-L. Wang, C.-Q. Song, and S.-P. Ji, Iterative solution to a class of complex matrix equations and its application in time-varying linear system, J. Appl. Math. Comput. 67 (2021), 317–341) algorithms, the proposed ones can make full use of the latest information and need less computations, which leads to higher computational efficiency. With the real representation of a complex matrix as a tool, we establish sufficient and necessary conditions for the convergence of the IMGI and the IMRGI algorithms. Finally, some numerical examples are given to illustrate the effectiveness and advantages of the proposed algorithms.

Similarity and consimilarity of hyper‐dual generalized quaternions

The aim of this paper is to investigate similarity and consimilarity of hyper‐dual generalized quaternions and their matrices. For this purpose, we give different conjugates according to the generalized quaternionic units . We present ‐consimilarity of hyper‐dual generalized quaternions and their matrices except hyper‐dual ‐quaternions. For the generalization consisting of hyper‐dual coefficients quaternion and split quaternion, we search ‐consimilarity and ‐consimilarity with the help of ‐conjugate and ‐conjugate. We also give ‐coneigenvalues and ‐coneigenvectors of the matrices of these generalizations. In addition, we examine right coneigenvalue problem in generalized quaternion matrices for real and split quaternions. The complex matrix representation obtained through the complex adjoint matrix representation of this generalization is introduced, and its properties are presented. Besides, we give algebraic methods for the concept of right coneigenvalues and coneigenvectors for matrices, which are the generalization of real quaternion and split quaternion.

Analytic and Structural Insights into Eigendecompositions of J2 Binormal Unitary Matrices on the Complex Plane

This research focuses on the eigende compositions of J2 binormal unitary matrices, denoted as J_2^bum (r). These matrices encapsulate matrix components with cohesive logical properties and secondary binormal behavior along a specified unit loop r∈J_2. Addressing gaps in current knowledge, we establish the existence of a decomposition J_2^bum (r)=Un(r)Bi(r)Un(r)^P, meeting conditions such that Un(r)^P=Un(r)^* (conjugate transpose), Bi(r) is a real binormal matrix, and both Un(r) and Bi(r) are analytic functions of a positive integer N. The decomposition extends the well-known Rellich theorem to matrix-valued functions exhibiting J2 binormal unitarity on the real number line. Furthermore, this theorem can be adapted for functions with analyticity and J2 binormal unitary characteristics along lines or circles in the complex plane, and the findings are extended to J2 binormal unitary matrices with elements satisfying a unitary series.

DM-GAN: CNN hybrid vits for training GANs under limited data

Generative adversarial network (GAN) training demands substantial data and computational resources. This paper aims to explore an economical approach for generating novel images with limited image data, addressing the challenge of data scarcity. Our contributions involve resolving the few-shot image generation challenge through the development of an unsupervised hybrid generative adversarial network named DM-GAN. We introduce a lightweight hybrid module (DC-Vit) comprising convolution and visual transformation, merging local and global features to enhance image perception, expressiveness, and ensure stable image generation. Additionally, a multi-scale adaptive skip connection module is incorporated to effectively mitigate the feature loss problem arising from inter-layer jumps, thereby producing more complete and regular images. To enhance the texture learning process and improve the quality and realism of synthesized images, we integrate the gray conjugate matrix into the loss function. Empirical evaluations are conducted on small sample datasets at various resolutions, including publicly accessible collections of art paintings, real-life photographs, and proprietary artifact image datasets. The experimental results unequivocally demonstrate the qualitative and quantitative superiority of our model over existing methods, underscoring its efficacy and robustness.

Parameters of the tensor tetraquark [formula omitted]

The mass and width of the tensor tetraquark T=bbc‾c‾ with spin-parity JP=2+ are calculated in the context of the QCD sum rule method. The tetraquark T is modeled as a diquark-antidiquark state built of components bTCγμb and c‾γνCc‾T with C being the charge conjugation matrix. The mass m=(12.795±0.095)GeV of the exotic tensor meson T is found by means of the two-point sum rule approach. Its full width Γ is evaluated by considering processes T→Bc−Bc−, Bc−Bc⁎−, and Bc⁎−Bc⁎−. Partial widths of these decays are computed by means of the three-point sum rule approach which is used to determine the strong couplings at relevant tetraquark-meson-meson vertices. Predictions obtained for the width ΓT=55.5−9.9+10.6MeV, as well as the mass of the tetraquark T can be useful in investigations of fully heavy four-quark mesons.

A modified gradient‐based iterative algorithm for solving the complex conjugate and transpose matrix equations

In this paper, we first develop the modified gradient‐based iterative (MGI) method for the complex conjugate and transpose matrix equations . By adopting the updated technique, we can make full use of the latest information to compute the next result, which leads to a faster convergence rate. In theory, we apply the real representation of a complex matrix and the vec‐operator to prove the convergence properties. Furthermore, we extend the MGI algorithm to solve the generalized complex conjugate and transpose matrix equations. Then, the necessary and sufficient conditions for convergence of the MGI algorithm are presented. Lastly, three numerical examples are introduced to testify the efficiency of our methods.

A novel structure-preserving algorithm for the singular value decomposition of biquaternion matrices

In this paper, we study the singular value decomposition of biquaternion matrices. We prove that the singular value decomposition of biquaternion matrices can be equivalently converted to the singular value decomposition of quaternion adjoint matrices. By means of the Cheng product, we propose a novel representation called Q -representation and design a Q structure-preserving algorithm to deal with the singular value decomposition of quaternion matrices, which makes use of high-level operations. The algorithm is more efficient than that in quaternion toolbox for matlab using quaternion arithmetics and real structure-preserving algorithm. The Q structure-preserving algorithm for singular value decomposition of biquaternion matrices is presented through the application of the Q structure-preserving algorithm for singular value decomposition of quaternion matrices. Numerical experiments are provided to demonstrate the efficiency of the Q structure-preserving algorithm. We apply the Q structure-preserving algorithm of quaternion singular value decomposition to improve the image denoising process.

On Generalizing the Method of Scarpis to Complex Hadamard Matrices

A complex Hadamard matrix is a matrix \(H_n \in {\{\omega^i | 1\leq i \leq m \}}^{n\times n}\) of order \(n\), where \(\omega\) is a primitive \(m^{th}\) root of unity, that satisfies \(H_n{H}^{*}_n=n{I_{n}}\), where \(H_n^{*}\) denotes the complex conjugate transpose of \(H_n\). We show that the Scarpis technique for constructing classic Hadamard matrices generalizes to Butson-type complex Hadamard matrices.

Vectorial holography over a multimode fiber.

Vectorial holography through a strongly scattering medium can facilitate various applications in optics and photonics. However, the realization of vectorial holography with arbitrary distribution of optical intensity is still limited because of experimental noise during the calibration of vectorial transmission matrix (TM) and reconstruction noise during the retrieval of input wavefront for a given holographic target. Herein, we propose and experimentally demonstrate the vectorial holography with arbitrary distribution of optical intensity over a multimode fiber (MMF) using the Tikhonov regularization. By optimizing the noise factor, the performance of vectorial holography over an MMF is improved compared with the conjugate transpose and inverse TM methods. Our results might shed new light on the optical communication and detection mediated by MMFs.

A criterion for the properness of star-transposition as an involution on n × n matrices over any associative star-ring

On the ring M n ( C ) of all n × n complex matrices A, the conjugate transpose involution A ↦ A * has the “properness” property A * A = 0 ⇒ A = 0 , as required for the existence of the partial order known as the “ * -order” on M n ( C ) . More generally, related questions are asked and answered about the ring M n ( R ) of all n × n matrices over an arbitrary associative ring R with any given involution R → R (as a generalization of the conjugacy map C → C ), yielding a corresponding “ * -transposition” involution on M n ( R ) . A criterion is found for this involution of M n ( R ) to be proper, so that M n ( R ) has a corresponding * -order. The special case R = Z k is also considered.

Modified and accelerated relaxed gradient-based iterative algorithms for the complex conjugate and transpose matrix equations

Modified and accelerated relaxed gradient-based iterative algorithms for the complex conjugate and transpose matrix equations

Exploring cone-shaped solitons, breather, and lump-forms solutions using the lie symmetry method and unified approach to a coupled breaking soliton model

In this research article, we investigate the coupled breaking soliton (cBS) model using two distinct analytical methods, namely, the Lie symmetry approach and the Unified method. We start by applying the Lie group technique to the cBS model, allowing us to establish infinitesimals, vector fields, commutative and adjoint tables, and an adjoint transformation matrix. Through the utilization of the adjoint transformation matrix, we identify a one-dimensional optimal system of subalgebras. This essential stage allows the cBS model to be reduced into several collections of ordinary differential equations (ODEs) relating to similarity variables resulting from symmetry reduction. By solving these ODE systems under specific parametric constraints, we successfully obtain exact solutions in terms of closed form. Furthermore, the Unified method is employed to address the governing equation, leading us to deduce polynomial and rational function solutions. The dynamic behaviours and characteristics of these such solutions are comprehensively explored through 3-dimensional (3D) plots and contour plots. The graphics show breather solitons, cone-shaped solitons, lump solitons, and patterns of flower petals, periodic solitons, and solitary waves. Additionally, we have connected our mathematical findings with real-world phenomena, which enrich our research work. Furthermore, breathers and lumps arise in many fields of mathematical physics, including fluid dynamics, plasma physics, ocean engineering, nonlinear optics, and physical sciences, as well as several other areas of nonlinear dynamics.

The Weighted, Relaxed Gradient-Based Iterative Algorithm for the Generalized Coupled Conjugate and Transpose Sylvester Matrix Equations

By applying the weighted relaxation technique to the gradient-based iterative (GI) algorithm and taking proper weighted combinations of the solutions, this paper proposes the weighted, relaxed gradient-based iterative (WRGI) algorithm to solve the generalized coupled conjugate and transpose Sylvester matrix equations. With the real representation of a complex matrix as a tool, the necessary and sufficient conditions for the convergence of the WRGI algorithm are determined. Also, some sufficient convergence conditions of the WRGI algorithm are presented. Moreover, the optimal step size and the corresponding optimal convergence factor of the WRGI algorithm are given. Lastly, some numerical examples are provided to demonstrate the effectiveness, feasibility and superiority of the proposed algorithm.

Linear codes invariant under cyclic endomorphisms

In this paper, we study linear codes invariant under a cyclic endomorphism [Formula: see text] called [Formula: see text]-cyclic codes. Since every cyclic endomorphism can be represented by a cyclic matrix with respect to a given basis, all of these matrices are similar. For simplicity we restrict ourselves to linear codes invariant under the right multiplication by a cyclic matrix [Formula: see text] that we call [Formula: see text]-cyclic codes, and when [Formula: see text] is the companion matrix [Formula: see text] of a given nonzero polynomial [Formula: see text] we call them [Formula: see text]-cyclic codes. The similarity relation between matrices helps us find connections between [Formula: see text]-cyclic codes and [Formula: see text]-cyclic codes, where [Formula: see text] is the minimal polynomial of [Formula: see text] The class of [Formula: see text]-cyclic codes contains cyclic codes and their various generalizations such as constacyclic codes, right and left polycyclic codes, monomial codes, and others. As common in the study of cyclic codes and their generalizations, we make use of the one-to-one correspondence between [Formula: see text]-cyclic codes and ideals of the polynomial ring [Formula: see text] where [Formula: see text] is the minimal polynomial of [Formula: see text] This correspondence leads to some basic characterizations of these codes such as generator and parity check polynomials among others. Next, we study the duality of these codes, where we show that the [Formula: see text]-dual of an [Formula: see text]-cyclic code is an [Formula: see text]-cyclic code, where [Formula: see text] is the adjoint matrix of [Formula: see text] with respect to [Formula: see text] and we explore some important results on the duality of these codes. Finally, we give examples as applications of some of the results and we construct some optimal codes.

MATRICES OF HYBRID NUMBERS

In this study, we investigate the matrices over the new extension of the real numbers in four dimensional space E2^4 called the hybrid numbers. Since the hybrid multiplication is noncommutative, this leads to finding a linear transformation on the complex field. Thus we characterize the hybrid matrices and examine their algebraic properties with respect to their complex adjoint matrices. Moreover, we define the co-determinant of hybrid matrices which plays an important role to construct the Lie groups.

Relaxation strategy for the block Landweber method

The block Landweber iteration is a general method for the solution of linear systems which is widely applied for image reconstructions. The convergence behavior of the block Landweber iteration is of both theoretical and practical importance. When the linear system is consistent, we derive a neat iterative representation and find an optimal relaxation coefficient by minimizing the 2-norm of the newly derived iteration matrix in the range of the conjugate transpose of the corresponding block matrix. We also establish the accelerating convergence relaxation strategies based on the maximum eigenvalue of each block matrix. When the linear system is inconsistent, we normalize each block matrix by the 2-norms of the each weighted block matrix at a uniform scale to give the relaxation strategies. The numerical simulations show that the relaxation strategies proposed for the block Landweber iteration in both consistent and inconsistent cases in this paper have the better reconstruction performances. Compared with the Landweber iteration, the block Landweber iteration has superiority in reconstruction results.

Numerical Simulation of the Conjugate Heat Transfer of a “Fluid–Solid Body” System on an Unmatched Grid Interface

Recently, when modeling transient problems of conjugate heat transfer, the independent construction of grid models for fluid and solid subdomains is increasingly being used. Such grid models, as a rule, are unmatched and require the development of special grid interfaces that match the heat fluxes at the interface. Currently, the most common sequential approach to modeling problems of conjugate heat transfer requires the iterative matching of boundary conditions, which can significantly slow down the process of the convergence of the solution in the case of modeling transient problems with fast processes. The present study is devoted to the development of a direct method for solving conjugate heat transfer problems on grid models consisting of inconsistent grid fragments on adjacent boundaries in which, in the general case, the number and location of nodes do not coincide. A conservative method for the discretization of the heat transfer equation by the direct method in the region of inconsistent interface boundaries between liquid and solid bodies is proposed. The proposed method for matching heat fluxes at mismatched boundaries is based on the principle of forming matched virtual boundaries, proposed in the GGI (General Grid Interface) method. A description of a numerical scheme is presented, which takes into account the different scales of cells and the sharply different thermophysical properties at the interface between liquid and solid media. An algorithm for constructing a conjugate matrix, the form of matrix coefficients responsible for conjugate heat transfer, and methods for calculating them are described. The operability of the presented method is demonstrated by the example of calculating conjugate heat transfer problems, the grid models of which consist of inconsistent grid fragments. The use of the direct conjugation method makes it possible to effectively solve both stationary and non-stationary problems using inconsistent meshes, without the need to modify them in the conjugation region within a single CFD solver.

Phase retrieval for affine groups over prime fields

We study phase retrieval for group frames arising from permutation representations, focusing on the action of the affine group of a finite field. We investigate various versions of the phase retrieval problem, including conjugate phase retrieval, sign retrieval, and matrix recovery. Our main result establishes that the canonical irreducible representation of the affine group Zp⋊Zp⁎ (with p prime), acting on the vectors in Cp with zero-sum, has the strongest retrieval property, allowing to reconstruct matrices from scalar products with a group orbit consisting of rank-one projections. We explicitly characterize the generating vectors that ensure this property, provide a linear matrix recovery algorithm and explicit examples of vectors that allow matrix recovery. We also comment on more general permutation representations.

Space-time POD and the Hankel matrix.

Time-delay embedding is an increasingly popular starting point for data-driven reduced-order modeling efforts. In particular, the singular value decomposition (SVD) of a block Hankel matrix formed from successive delay embeddings of the state of a dynamical system lies at the heart of several popular reduced-order modeling methods. In this paper, we show that the left singular vectors of this Hankel matrix are a discrete approximation of space-time proper orthogonal decomposition (POD) modes, and the singular values are square roots of the POD energies. Analogous to the connection between the SVD of a data matrix of snapshots and space-only POD, this connection establishes a clear interpretation of the Hankel modes grounded in classical theory, and we gain insights into the Hankel modes by instead analyzing the equivalent discrete space-time POD modes in terms of the correlation matrix formed by multiplying the Hankel matrix by its conjugate transpose. These insights include the distinct meaning of rows and columns, the implied norm in which the modes are optimal, the impact of the time step between snapshots on the modes, and an interpretation of the embedding dimension/height of the Hankel matrix in terms of the time window on which the modes are optimal. Moreover, the connections we establish offer opportunities to improve the convergence and computation time in certain practical cases, and to improve the accuracy of the modes with the same data. Finally, popular variants of POD, namely the standard space-only POD and spectral POD, are recovered in the limits that snapshots used to form each column of the Hankel matrix represent flow evolution over short and long times, respectively.