Symmetry Of Matrix Research Articles

Permutable Symmetric Hadamard Matrices in Quaternion Algebra and Engineering Applications

In this paper, aiming to develop the group and out-of-group formalization of the symmetry concept, the preservation of a matrix symmetry after row permutation is considered by the example of the maximally permutable \emph{normalized} Hadamard matrices which row and column elements are either plus or minus one. These matrices are used to extend the additive decomposition of a linear operator into symmetric and skew-symmetric parts using several commuting operations of the Hermitian conjugation type, for the quaternionic generalization of a vector cross product, as well as for creating educational puzzles and other applications.

Investigating the formation mechanism of void lattice in tungsten under neutron irradiation: from collision cascades to ordered nanovoids

The “void lattice” formed by periodic arrangements of voids, usually replicating the symmetry and crystallographic orientation of the host matrix, is an interesting phenomenon in materials under neutron irradiation. In this work, taking tungsten (W) as an example, we explore the formation mechanism of the void lattice using an object kinetic Monte Carlo (OKMC) model together with the collision cascades simulated by the molecular dynamics method. The entire formation processes from the chaotic neutron irradiation defects to the observable void lattice are reproduced via OKMC simulation, which could be divided into three stages: nucleation, incubation and growth. It is found that both the one-dimensional (1D) migration of SIAs and the fraction of clustered vacancies in cascades play a critical role in the formation of the void lattice. On the one hand, the 1D migration of SIAs leads to the mutual protection of voids aligned in <111> directions. The self-shielded voids may therefore grow faster than the unaligned ones. On the other hand, a moderate fraction of clustered vacancies in cascades guarantees the stable nucleation and growth rates of voids. Once the density of the <111> aligned voids reaches a critical value, the shrinkage rate of the unaligned voids will overwhelm their growth rate, leading to the formation of void-free channels and thus the void lattice. Our results reveal the synergistic effects of the fraction of clustered vacancies in cascades and 1D migration of SIAs for the formation of the void lattice in W under neutron irradiation, which improves the fundamental understanding of the self-assembled microstructures in irradiated materials.

Set-theoretic Yang\u2013Baxter & reflection equations and quantum group symmetries

Connections between set-theoretic Yang–Baxter and reflection equations and quantum integrable systems are investigated. We show that set-theoretic R-matrices are expressed as twists of known solutions. We then focus on reflection and twisted algebras and we derive the associated defining algebra relations for R-matrices being Baxterized solutions of the A-type Hecke algebra {mathcal {H}}_N(q=1). We show in the case of the reflection algebra that there exists a “boundary” finite sub-algebra for some special choice of “boundary” elements of the B-type Hecke algebra {mathcal {B}}_N(q=1, Q). We also show the key proposition that the associated double row transfer matrix is essentially expressed in terms of the elements of the B-type Hecke algebra. This is one of the fundamental results of this investigation together with the proof of the duality between the boundary finite subalgebra and the B-type Hecke algebra. These are universal statements that largely generalize previous relevant findings and also allow the investigation of the symmetries of the double row transfer matrix.

Determination of the elastic constants of thermally modified beech by ultrasound and static tests coupled with 3D digital image correlation

Fagus sylvatica L. (European beech) is one of the most widespread hardwood species growing in Europe, which is currently undergoing of in-depth research for the development of engineering products to use its excellent mechanical properties. As its natural durability is low, heat treatment is investigated as a means to enhance its biological durability, as well as its dimensional stability. Reliable models with a full material description including the elastic constants are necessary for material and structural modelling and design. The aim of this work was to comprehensively characterise European beech subjected to three different intensities of heat treatments. It is described as an orthotropic material by determining all of the independent elastic constants: three Young’s moduli, three shear moduli and six Poisson’s ratios. Both static (by compression) and dynamic (by ultrasound) experimental methods were considered for comparison purposes. The compression tests were coupled with 3D digital image correlation (DIC) technique to perform optical full-field analyses of strains. Characterization of untreated beech was also carried out and compared with literature values. The usual assumption of symmetry of the compliance matrix was verified. The results confirmed that heat treatment influenced the elastic behaviour of the material. However, the impact of the treatment differed among the elastic components, with non-uniform trends with the intensity of the heat treatments.

Frequency domain spectral element method for modelling poroelastic waves in 3-D anisotropic, heterogeneous and attenuative porous media

SUMMARY Simulating poroelastic waves in large-scale 3-D problems having porous media coupled with elastic solids and fluids is computationally challenging for traditional methods. It is well established that the spectral element method (SEM) is more effective than the traditional methods like the finite element method (FEM) when dealing with complex geophysical problems, for its high-order accuracy with exponential convergence. However, at present, little research has been done for SEM in the frequency domain, which will be more efficient than the time-domain SEM for narrowband simulations with multiple sources, material dispersion and attenuation. Herein, we systematically develop a SEM in the frequency domain to simulate coupled poroelastic, elastic and acoustic waves in anisotropic (i.e. porosity, permeability and elastic coefficients with anisotropy), heterogeneous, and lossy media. Furthermore, we completely remove the dimension inconsistency between the displacement field and the pressure in porous media to reduce the condition number of the system matrix by around 16 orders of magnitude while maintaining the symmetry of the system matrix. To solve the multiphysics coupling problems, we apply different coupling conditions to different interface types, and use basis functions to discretize the corresponding governing equations. Numerical examples show that the proposed SEM can obtain higher accuracy with much fewer unknowns compared with the FEM and has the capacity to solve the large-scale real coupling problems.

Many-Body Delocalization as Symmetry Breaking.

We present a framework in which the transition between a many-body localized (MBL) phase and an ergodic one is symmetry breaking. We consider random Floquet spin chains, expressing their averaged spectral form factor (SFF) as a function of time in terms of a transfer matrix that acts in the space direction. The SFF is determined by the leading eigenvalues of this transfer matrix. In the MBL phase, the leading eigenvalue is unique, as in a symmetry-unbroken phase, while in the ergodic phase and at late times the leading eigenvalues are asymptotically degenerate, as in a system with degenerate symmetry-breaking phases. We identify the broken symmetry of the transfer matrix, introduce a local order parameter for the transition, and show that the associated correlation functions are long-ranged only in the ergodic phase.

The determination of redundant test patterns based on binary matrix symmetry properties

The object of research are digital circuit control tests. The model of a digital circuit is of a grey-box type. Test patterns (impacts) are combinations of logical zeros and ones, applied to circuit inputs. A binary matrix is the response of a digital circuit to input impacts. The examined faults are of «bridging faults» and «stuck-at faults» types. We distinguish the structural units of a test, as well as transformations, maintaining its structure. The measure of binary matrix symmetry is defined. Based on the measure obtained, the criteria are introduced, which help to calculate the utility of another test impact upon a digital circuit. To investigate the criteria, both pseudo-random and deterministic test sequences have been used. Among others, we have observed such algorithmic sequences as a sliding-ONE test, a logarithmic test, a galloping test pattern and a checkerboard test. The proposed approach gives the possibility to identify and exclude the impacts, not carrying useful information, from a test of combinational circuit control. The developed criteria can also be applied to construct memory tests. The analysis of experimental data allows us to make a conclusion about the advisability of using the proposed mathematical apparatus in the theory and practice of digital device testing, as well as in processes of test development.

Anomalous thermodynamics in homogenized generalized Langevin systems

We study functionals, such as heat and work, along trajectories of a class of multi-dimensional generalized Langevin systems in various limiting situations that correspond to different level of homogenization. These are the situations where one or more of the inertial time scale(s), the memory time scale(s) and the noise correlation time scale(s) of the systems are taken to zero. We find that, unless one restricts to special situations that do not break symmetry of the Onsager matrix associated with the fast dynamics, it is generally not possible to express the effective evolution of these functionals solely in terms of trajectory of the homogenized process describing the system dynamics via the widely adopted Stratonovich convention. In fact, an anomalous term is often needed for a complete description, implying that convergence of these functionals needs more information than simply the limit of the dynamical process. We trace the origin of such impossibility to area anomaly, thereby linking the symmetry breaking and area anomaly. This hold important consequences for many nonequilibrium systems that can be modeled by generalized Langevin equations. Our convergence results hold in a strong pathwise sense.

Two- and three-dimensional size-dependent couple stress response using a displacement-based variational method

Based upon the principle of minimum total potential energy for consistent couple stress theory, a Ritz variational approach is developed using tensor product B-splines for both two- and three-dimensional elastostatic analysis. The underlying theory is size-dependent due to incorporation of the energy conjugate skew-symmetric couple-stress and mean curvature tensors, in addition to the force-stress and strain conjugate pair of classical theory. The use of B-splines as the basis functions can assure the required C1 continuity of the displacement field, while also permitting higher order representations. Both displacements and rotations are essential boundary conditions in this theory. Displacements can be enforced in the usual way, but rotations require special treatment to maintain symmetry and positive definiteness of the stiffness matrix for well-posed elastostatic problems. Several computational examples are considered to validate the formulation, illustrate convergence characteristics, and investigate mechanical behavior under consistent couple stress theory. All previous numerical analysis of consistent couple stress theory has been limited to plane strain problems. Thus, the extension here to three-dimensions is of great importance, especially because the behavior in several cases shows significant deviation from the plane strain solutions. Consequently, the variational formulations and computational methodology presented in this paper can play a critical role in assessing the predictive capability of consistent couple stress theory and in understanding size-dependent elastic response.

μ − τ reflection symmetry in the standard parametrization and contributions from charged lepton sector

The μ−τ reflection symmetry of the lepton mixing matrix accommodates maximal atmospheric mixing (θ23=π/4) as well as maximal Dirac CP phase (δ=±π/2) for the Dirac case. In the standard parametrization of the PMNS matrix the reflection symmetric nature is not directly visible while substituting the maximal values of the mixing parameters. This issue has been addressed in this paper. It is found that the reflection symmetry in the ‘standard’ PMNS matrix can be restored by allowing maximal values of the Majorana CP phases (α, β) as well, along with maximal δ. To accommodate non-maximal values of θ23 and δ we consider charged lepton contributions to the neutrino mixing and implement the proposed scheme of reflection symmetry in the neutrino mixing matrix. The charged lepton correction scheme succeeds in the prediction of lepton mixing parameters consistent with the global 3ν oscillation data.

A Parallel Jacobi-Embedded Gauss-Seidel Method

A broad range of scientific simulations involve solving large-scale computationally expensive linear systems of equations. Iterative solvers are typically preferred over direct methods when it comes to large systems due to their lower memory requirements and shorter execution times. However, selecting the appropriate iterative solver is problem-specific and dependent on the type and symmetry of the coefficient matrix. Gauss-Seidel (GS) is an iterative method for solving linear systems that are either strictly diagonally dominant or symmetric positive definite. This technique is an improved version of Jacobi and typically converges in fewer iterations. However, the sequential nature of this algorithm complicates the parallel extraction. In fact, most parallel derivatives of GS rely on the sparsity pattern of the coefficient matrix and require matrix reordering or domain decomposition. In this article, we introduce a new algorithm that exploits the convergence property of GS and adapts the parallel structure of Jacobi. The proposed method works for both dense and sparse systems and is straightforward to implement. We have examined the performance of our method on multicore and many-core architectures. Experimental results demonstrate the superior performance of the proposed algorithm compared with GS and Jacobi. Additionally, performance comparison with built-in Krylov solvers in MATLAB showed that in terms of time per iteration, Krylov methods perform faster on CPUs, but our approach is significantly better when executed on GPUs. Lastly, we apply our method to solve the power flow problem, and the results indicate a significant improvement in runtime, reaching up to 87 times faster speed compared with GS.

When do cross-diffusion systems have an entropy structure?

In this note, necessary and sufficient conditions for the existence of an entropy structure for certain classes of cross-diffusion systems with diffusion matrix A(u) are given, based on results from matrix factorization. The entropy structure is important in the analysis for such equations since A(u) is typically neither symmetric nor positive definite. In particular, the normal ellipticity of A(u) for all u and the symmetry of the Onsager matrix implies its positive definiteness and hence an entropy structure. If A is constant or constant up to nonlinear perturbations, the existence of an entropy structure is equivalent to the normal ellipticity of A. The results are applied to various examples from physics and biology. Finally, the normal ellipticity of the n-species population model of Shigesada, Kawasaki, and Teramoto is proved.

Understanding the Formation Mechanism of Void Lattice Under Irradiation: From Collision Cascades to Self-Assembled Nanovoids

The formation of periodic arrangements of voids is an interesting phenomenon occurring in materials under neutron irradiation, usually replicating the symmetry and crystallographic orientation of the host matrix, hence called “void lattice”. Here, taking tungsten as an example, we explore the formation mechanism of the void lattice using an object kinetic Monte Carlo (OKMC) model through the collision cascades simulated using molecular dynamics method. Specifically, the detailed processes from the chaotic neutron irradiation defects to the observable void lattice are obtained via OKMC simulation, which is a prerequisite for understanding its formation mechanism. The formation of the void lattice could be divided into three stages: nucleation, incubation and growth. Both the one-dimensional (1D) migration of SIAs and the fraction of clustered vacancies in cascades play a critical role in the emergence of the void lattice. On the one hand, the 1D migration of SIAs leads to the mutual protection of voids aligned in directions. The self-shielded voids may therefore grow faster compared to the unaligned ones. On the other hand, a moderate fraction of clustered vacancies in cascades guarantees a stable rate for both nucleation and growth of voids. Once the density of the aligned voids reaches a critical value, the dissolution rate of the unaligned voids will overwhelm their nucleation rate, leading to the formation of void-free channels and thus the void lattice. Our results reveal the interrelated mechanism of the 1D migration of SIAs and intra-cascade vacancy clustering for the formation of the void lattice under neutron irradiation, which improves our fundamental understanding of the self-assembled microstructures in irradiated materials.

A Note on the Periodic Solutions for a Class of Third Order Differential Equations

The aim of the present work is to study the necessary and sufficient conditions for the existence of periodic solutions for a class of third order differential equations by using the averaging theory. Moreover, we use the symmetry of the Monodromy matrix to study the stability of these solutions.

A basic orthotropic viscoelastic model for composite and wood materials considering available experimental data and time-dependent Poisson’s ratios

Long-term deformation in creep is of significant engineering importance. For anisotropic materials, such as wood, composites and reinforced concrete, creep testing in several axial directions including shear is necessary to obtain a creep model which is able to predict deformation in the basic orthotropic case. Such a full set of experimental data is generally not available, and simplifying assumptions are typically made to conceive a useful 3D model. These assumptions should preferably be made based on the material behaviour and sound engineering arguments. This problem appears to be addressed in many different ways and sometimes the assumptions are not well justified. In the present study, we examine 3D creep of wood and composite materials. Particular emphasis is made on explaining the choices made in developing the model, considering practicality, incomplete material data and the specific behaviour of wood and composites. An orthotropic linear viscoelastic model is implemented as a material model in a commercial FE software. The constitutive equations are derived in the 1D case using a hereditary approach, then later generalized to the 3D formulation. Guidelines are shown how to implement it into the FE software to predict creep of components and structures. Although the model itself is conventional, the effect of considering time-dependent Poisson’s ratios is investigated here, as well an optimization approach when inserting inevitably asymmetric experimental creep data into the model. As far as the authors know, creep of wooden materials have not been defined using this approach before. The model of interest is calibrated against experimental data. Examples using experimental results from solid wood data and a unidirectional fiber composite are demonstrated. The results show that the model is able to capture the orthotropic behaviour adequately. Orthotropy requires symmetry of the creep compliance matrix, which typically is not the case experimentally. It is shown that in rendering the matrix symmetric, one needs to decide which direction is more important. It is also shown that the frequently employed assumption of constant Poisson’s ratios should be made with caution.

Collision rate ansatz for quantum integrable systems

For quantum integrable systems the currents averaged with respect to a generalized Gibbs ensemble are revisited. An exact formula is known, which we call ``collision rate ansatz". While there is considerable work to confirm this ansatz in various models, our approach uses the symmetry of the current-charge susceptibility matrix, which holds in great generality. Besides some technical assumptions, the main input is the availability of a self-conserved current, i.e. some current which is itself conserved. The collision rate ansatz is then derived. The argument is carried out in detail for the Lieb-Liniger model and the Heisenberg XXZ chain. It is also explained how from the existence of a boost operator a self-conserved current can be deduced.

Asymmetric tri-bi-maximal mixing and residual symmetries

Asymmetric tri-bi-maximal mixing is a recently proposed, grand unified theory (GUT) based, flavor mixing scheme. In it, the charged lepton mixing is fixed by the GUT connection to down-type quarks and a [Formula: see text] flavor symmetry, while neutrino mixing is assumed to be tri-bi-maximal (TBM) with one additional free phase. Here we show that this additional free phase can be fixed by the residual flavor and CP symmetries of the effective neutrino mass matrix. We discuss how those residual symmetries can be unified with [Formula: see text] and identify the smallest possible unified flavor symmetries, namely [Formula: see text] and [Formula: see text]. Sharp predictions are obtained for lepton mixing angles, CP violating phases and neutrinoless double beta decay.

State estimation based on improved cubature Kalman filter algorithm

During the recursive calculate process of the cubature Kalman filter (CKF), the covariance matrix tends to lose positive definiteness and noise statistical characteristics are inaccurate, which results in inaccurate filtering or even filter divergence. This study presents an improved algorithm based on the CKF. The algorithm combines the square root filter algorithm and the Sage–Husa maximum a posterior noise estimator, which can ensure the non-negative determination and symmetry of the covariance matrix and has the ability to deal with unknown and time-varying noise statistical characteristics in the filtering process adaptively. In the multi-dimension system, the noise covariance matrix may dissatisfy non-negative definiteness and result in filter divergence, and then the noise covariance matrix estimator is improved. The analysis is verified by state estimation example of the non-linear system, compared with the standard CKF, the accuracy of the adaptive square root CKF (ASRCKF) state estimation is increased by 63.13, 63.88, and 42.71%, respectively. Finally, the effectiveness of the ASRCKF is verified by the state estimation of the permanent magnet linear synchronous motor.

Indirect-Inversion Algorithm via Precise Integration for Ill-Conditioned Matrix in Ambiguity Resolution

Global navigation satellite system (GNSS) positioning depends on the correct integer ambiguity resolution (AR). If the double difference equation for solving the float solution remains ill-conditioned, often happening due to the environment complexity and the equipment mobility, the correct AR is difficult to achieve. Concerning the ill-conditioned problem, methods of modifying the equation coefficient matrix are widely applied, whose effects are heavily dependent on modifying parameters. Besides, the direct-inversion of the ill-conditioned coefficient matrix can lead to a reduction in the accuracy and stability of the float solution. To solve the problem of ill-conditioned matrix inversion and further improve the accuracy, the present study for the first time proves the positive definite symmetry of the coefficient matrix in AR model and employs precise integration method to the indirect inverse of coefficient matrix. AR model for the GNSS positioning and the general resolving strategies introduction are briefly introduced. An indirect-inversion algorithm via precise integration for ill-conditioned coefficient matrix is proposed. According to the simulations and comparisons, the proposed strategy has higher precision and stability on float solution, and less dependence on modifying parameters.

Information geometry in quantum field theory: lessons from simple examples

Motivated by the increasing connections between information theory and high-energy physics, particularly in the context of the AdS/CFT correspondence, we explore the information geometry associated to a variety of simple systems. By studying their Fisher metrics, we derive some general lessons that may have important implications for the application of information geometry in holography. We begin by demonstrating that the symmetries of the physical theory under study play a strong role in the resulting geometry, and that the appearance of an AdS metric is a relatively general feature. We then investigate what information the Fisher metric retains about the physics of the underlying theory by studying the geometry for both the classical 2d Ising model and the corresponding 1d free fermion theory, and find that the curvature diverges precisely at the phase transition on both sides. We discuss the differences that result from placing a metric on the space of theories vs.~states, using the example of coherent free fermion states. We compare the latter to the metric on the space of coherent free boson states and show that in both cases the metric is determined by the symmetries of the corresponding density matrix. We also clarify some misconceptions in the literature pertaining to different notions of flatness associated to metric and non-metric connections, with implications for how one interprets the curvature of the geometry. Our results indicate that in general, caution is needed when connecting the AdS geometry arising from certain models with the AdS/CFT correspondence, and seek to provide a useful collection of guidelines for future progress in this exciting area.