Block-diagonal Research Articles

Quantitative evaluation of the degradation amount of the silane coupling layer of CAD/CAM resin composites by water absorption.

Degradation of silane coupling layers by water ingress in computer-aided design/computer-aided manufacturing (CAD/CAM) of resin composites has been reported qualitatively. In this study, we quantitatively evaluated how water absorption of CAD/CAM resin composites affects the silane coupling layer by in vitro and in silico methods. A Katana Avencia block (KAB) and an experimental matrix block composed of only a matrix resin were used to evaluate the effect of water immersion for seven days on the elastic modulus. X-ray photoelectron spectroscopy (XPS) with fluorine-labeling of the KAB was performed to evaluate the atomic percentage of F1s, which represents the hydrolysis amount by water immersion. In silico analysis of the three-dimensional model of the KAB was performed to determine the coupling ratios before and after water immersion. The elastic modulus of the KAB was 8.2 GPa before and 6.9 GPa after immersion in water. The atomic percentages of F1s in the after- and before-immersion groups were 14.31% and 11.52%, respectively, suggesting that hydrolysis of the silane coupling layer occurred during water immersion. From in silico analysis of the three-dimensional model of the KAB, the coupling ratio was predicted to be 78.2% before water immersion. After water immersion, the coupling ratio was predicted to be 68.4%. The in vitro and in silico approaches established in this study were able to predict the silane coupling ratios of CAD/CAM resin composites, and they showed that the silane coupling ratio decreased by water absorption.

ON PRESSURE WAVES IN SATURATED FRACTURED POROUS MEDIUM WITH DAMAGEABLE MATRIX

A double-porosity model of single-phase flow induced by depression in fractured porous medium with damageable matrix is proposed. The development of small-scale fracturing leads to an increase in the permeability of matrix blocks and the intensification of mass transfer between subsystems of the double-porosity medium. By analyzing the inequality of dissipation, the thermodynamically consistent governing relations and the equation for evolution of the damage parameter in the matrix are derived. For the obtained system of equations, an initial-boundary value problem is formulated and solved numerically that models coupled processes of fluid flow, fracture, and changes in the stress-strain state in a loaded half-space with double porosity, which was initially in equilibrium under abnormally high pore pressure. Development of the damaged zone accompanying the pressure wave is analyzed. Two types of damage zone development are shown to exist. Choice between these solutions depends on the parameters of the medium.

Discretization of the Functionals With Prescribed Derivative for Linear Systems With Multiple Delays

This work aims to derive a sufficient stability condition for linear systems with multiple delays based on a discretization procedure of Lyapunov – Krasovskii functionals. This procedure relies on a piecewise constant approximation of matrix kernels of the functionals with prescribed derivatives. The approximation error is explicitly bounded, and the resulting stability test is expressed in terms of nonnegative definiteness of a block matrix that depends on the values of delay Lyapunov matrix at discretization points and the system matrices.

Further characterizations and representations of the Minkowski inverse in Minkowski space

<abstract><p>This paper serves to identify some new characterizations and representations of the Minkowski inverse in Minkowski space. First of all, a few representations of $ \{1, 3^{\mathfrak{m}}\} $-, $ \{1, 2, 3^{\mathfrak{m}}\} $-, $ \{1, 4^{\mathfrak{m}}\} $- and $ \{1, 2, 4^{\mathfrak{m}}\} $-inverses are given in order to represent the Minkowski inverse. Second, some famous characterizations of the Moore-Penrose inverse are extended to that of the Minkowski inverse. Third, using the Hartwig-Spindelböck decomposition, we present a representation of the Minkowski inverse. And, based on this result, an interesting characterization of the Minkowski inverse is showed by a rank equation. Finally, we obtain several new representations of the Minkowski inverse in a more general form, by which the Minkowski inverse of a class of block matrices is given.</p></abstract>

A study of the equivalence of inference results in the contexts of true and misspecified multivariate general linear models

<abstract><p>In practical applications of regression models, we may meet with the situation where a true model is misspecified in some other forms due to certain unforeseeable reasons, so that estimation and statistical inference results obtained under the true and misspecified regression models are not necessarily the same, and therefore, it is necessary to compare these results and to establish certain links between them for the purpose of reasonably explaining and utilizing the misspecified regression models. In this paper, we propose and investigate some comparison and equivalence analysis problems on estimations and predictions under true and misspecified multivariate general linear models. We first give the derivations and presentations of the best linear unbiased predictors (BLUPs) and the best linear unbiased estimators (BLUEs) of unknown parametric matrices under a true multivariate general linear model and its misspecified form. We then derive a variety of necessary and sufficient conditions for the BLUPs/BLUEs under the two competing models to be equal using a series of highly-selective formulas and facts associated with ranks, ranges and generalized inverses of matrices, as well as block matrix operations.</p></abstract>

Generalized numerical radius inequalities involving positive semidefinite block matrices

Generalized numerical radius inequalities involving positive semidefinite block matrices

Dual Drazin inverses of dual matrices and dual Drazin-inverse solutions of systems of linear dual equations

In this paper, we study a kind of dual generalized inverse, which is called the dual Drazin inverse. Unlike the real matrices case, the dual Drazin inverse of a square dual matrix may not exist. It is shown that the dual Drazin inverse is unique when it exists. Some necessary and sufficient conditions for the existence of the dual Drazin inverse are presented. A compact formula for the computation of the dual Drazin inverse is given when it exists. Moreover, we find an unexpected result that the dual Drazin inverse can be obtained by computing the Drazin inverse of a 2 ? 2 upper triangular block matrix. We also introduce the dual Drazin-inverse solution of systems of linear dual equations. Some characterizations of the dual Drazin-inverse solution are given. In addition, some numerical examples are provided to illustrate the results.

A note on the Moore-Penrose inverse of block matrices

Motivated by the representation for the Moore-Penrose inverse of the block matrix over a *-regular ring presented in [R.E. Hartwig and P. Patr?cio, When does the Moore-Penrose inverse flip? Operators and Matrices, 6(1):181-192, 2012], we show that the formula of the Moore-Penrose inverse is the same as the expression given by [Nieves Castro-Gonz?lez, Jianlong Chen and Long Wang, Further results on generalized inverses in rings with involution, Elect. J. Linear Algebra, 30:118-134, 2015].

Modeling of Nonlinear Control System with Prior Knowledges based on Koopman Operator

Recently, extended dynamic mode decomposition (EDMD) has become a popular tool for system identification. The EDMD is a data-driven algorithm to identify a linear or bilinear predictor of the nonlinear system based on the Koopman operator. Although linear and bilinear models are suitable for real-time control, these models’ prediction accuracy may not be sufficient for problems such as trajectory planning that require long-term prediction. To address this issue, we introduced the nonlinear predictor obtained from a block matrix representation of the bilinear predictor and compared it with the bilinear model. Furthermore, identifying a physically reasonable predictor from the limited number of noisy experimental data is also a challenging problem. In this study, we utilized several simple prior knowledges to achieve stability and sparsity and to identify the model that has correct dependencies between state and control input. We identified the forklift dynamics from real experimental data and showed that the obtained model has high prediction accuracy and is suitable for optimal trajectory planning.

Some generaliztions of numerical radius inequalities including the off-diagonal parts of block matrices

Some generaliztions of numerical radius inequalities including the off-diagonal parts of block matrices

Statistical inference of a stochastically restricted linear mixed model

<abstract><p>This article compares a predictor with the best linear unbiased predictor (BLUP) for a unified form of all unknown parameters under a stochastically restricted linear mixed model (SRLMM) in terms of the mean squared error matrix (MSEM) criterion. The methodology of block matrix inertias and ranks is employed to compare the MSEMs of these predictors. The comparison results are also demonstrated for a linear mixed model with and without an exact restriction, as well as special cases of the unified form of all unknown parameters in the SRLMM.</p></abstract>

Equivalence on Hiroshima's type inequalities for positive semidefinite block matrices

Equivalence on Hiroshima's type inequalities for positive semidefinite block matrices

Some norm inequalities for some positive block matrices

Some norm inequalities for some positive block matrices

Characterizations of matrix equalities involving the sums and products of multiple matrices and their generalized inverse

<abstract><p>It is common knowledge that matrix equalities involving ordinary algebraic operations of inverses or generalized inverses of given matrices can be constructed arbitrarily from theoretical and applied points of view because of the noncommutativity of the matrix algebra and singularity of given matrices. Two of such matrix equality examples are given by $ A_1B_1^{g_1}C_1 + A_2B_2^{g_2}C_2 + \cdots + A_kB_k^{g_k}C_{k} = D $ and $ A_1B_1^{g_1}A_2B_2^{g_2} \cdots A_kB_k^{g_k}A_{k+1} = A $, where $ A_1 $, $ A_2 $, $ \ldots $, $ A_{k+1} $, $ C_1 $, $ C_2 $, $ \ldots $, $ C_{k} $ and $ A $ and $ D $ are given, and $ B_1^{g_1} $, $ B_2^{g_2} $, $ \ldots $, $ B_k^{g_k} $ are generalized inverses of matrices $ B_1 $, $ B_2 $, $ \ldots $, $ B_k $. These two matrix equalities include many concrete cases for different choices of the generalized inverses, and they have been attractive research topics in the area of generalized inverse theory. As an ongoing investigation of this subject, the present author presents in this article several groups of new results and facts on constructing and characterizing the above matrix equalities for the mixed combinations of $ \{1\} $- and $ \{1, 2\} $-generalized inverses of matrices with $ k = 2, 3 $ by using some elementary methods, including a series of explicit rank equalities for block matrices.</p></abstract>

Unsupervised Functional Data Clustering Based on Adaptive Weights

In recent years, functional data has been widely used in finance, medicine, biology and other fields. The current clustering analysis can solve the problems in finite-dimensional space, but it is difficult to be directly used for the clustering of functional data. In this paper, we propose a new unsupervised clustering algorithm based on adaptive weights. In the absence of initialization parameter, we use entropy-type penalty terms and fuzzy partition matrix to find the optimal number of clusters. At the same time, we introduce a measure based on adaptive weights to reflect the difference in information content between different clustering metrics. Simulation experiments show that the proposed algorithm has higher purity than some algorithms.

Some New Properties on Block Matrices Using MATLAB Code

In this article, we study some new properties of the block matrices. Introduce the methods for finding the block matrices of any matrix. We prove some of these methods by using the matrix analysis methods. Some results of this paper are proved by using the permutations of matrices of the block matrices. In the last of this study, some examples are introduced to explain the results of this work and we give some MATLAB code in the appendix to explain the results of this article.

Entropy generation analysis of MHD hybrid nanofluid flow due to radiation with non-erratic slot-wise mass transfer over a rotating sphere

Flow over rotating spheres has serious applications in fiber coating, rotating machinery design and parts, and projectile missions. Therefore, the current study investigates the flow and entropy generation of a radiative magneto-hybrid nanofluid flow over rotating sphere. Further, the temperature-sensorial water properties in studying water-based nanofluids are ignored, though this erratic behavior of water characteristics influences the physical characteristics of the corresponding hybrid nanofluid. Hence the current framework is one of the foremost projects to introduce the variable nature of water properties in the water-hosted hybrid nanofluid flow analysis. The investigation is accomplished in the account of the modified Buongiorno model (MBM). The flow separation in this kind of flow geometry is controlled by using non-erratic slot-mass transfer. The mathematical representations of the physical principles of the flow are solved using (i) congenial transformation (ii) quasilinearization (iii) methods of finite differences to form a block matrix system, and (iv) Varga’s iterative algorithm. Some of the major outcomes are: The flow separation can be significantly delayed using slot suctions, precisely appointing them downstream with higher strengths; heat transport performance drastically subsidies for high viscous dissipation (Ec), entropy produces at higher rate for viscous heating (Br) and high angular speed (Ω).

The "Elliptic" matrices and a new kind of cryptography

In [17], A. chillali et al introduce a new cryptographic method based on matrices over a finite field Fpn , where p is a prime number. In this paper, we will generate this method in a new group of square block matrices based on an elliptic curve, called "elliptic" matrices.

Eigenvalue bounds for double saddle-point systems

Abstract We derive bounds on the eigenvalues of a generic form of double saddle-point matrices. The bounds are expressed in terms of extremal eigenvalues and singular values of the associated block matrices. Inertia and algebraic multiplicity of eigenvalues are considered as well. The analysis includes bounds for preconditioned matrices based on block diagonal preconditioners using Schur complements, and it is shown that in this case the eigenvalues are clustered within a few intervals bounded away from zero. Analysis for approximations of Schur complements is included. Some numerical experiments validate our analytical findings.

Miscellaneous results related to the Gaussian product inequality conjecture for the joint distribution of traces of Wishart matrices

This note reports partial results related to the Gaussian product inequality (GPI) conjecture for the joint distribution of traces of Wishart matrices. In particular, several GPI-related results from Wei [32] and Liu et al. [15] are extended in two ways: by replacing the power functions with more general classes of functions, and by replacing the usual Gaussian and multivariate gamma distributional assumptions by the more general trace-Wishart distribution assumption. These findings suggest that a Kronecker product form of the GPI holds for diagonal blocks of any Wishart distribution.