Fundamental Matrix Research Articles

Polarization clustering of biological structures with Mueller matrix parameters.

Mueller matrix imaging polarimetry (MMIP) is a promising technique for the characterization of biological tissues, including the classification of microstructures in pathological diagnosis. To expand the parameter space of Mueller matrix parameters, we propose new vector parameters (VPs) according to the Mueller matrix polar decomposition method. We measure invasive bladder cancer (IBC) with extensive necrosis and high-grade ductal carcinoma in situ (DCIS) with MMIP, and the regions of cancer cells and fibrotic stroma are classified with the VPs. Then the proposed and existing VPs are mapped on the Poincaré sphere with 3D visualization, and an indicator of spatial feature is defined based on the minimum enclosing sphere to evaluate the classification capability of the VPs. For both IBC and DCIS, the results show that the proposed VPs exhibit evident contrast between the regions of cancer cells and fibrotic stroma. This study broadens the fundamental Mueller matrix parameters and helps to improve the characterization ability of the MMIP technique.

Linear quaternion-valued difference equations: Representation of solutions, controllability, and observability

In this paper, we present the fundamental theory of linear quaternion-valued difference equations. Firstly, we derive general solutions for linear homogeneous equations and give the algorithm for calculating the fundamental matrix in the case of the diagonalizable form and Jordan form. Secondly, we apply the variation of the constant formula and Z transformation to study general solutions of linear nonhomogeneous equations. We obtain the representation of solutions in the case of quaternion and complex numbers. Thirdly, we adopt the ideas from the Gram matrix and the rank of the criteria to establish sufficient and necessary conditions to guarantee that linear quaternion-valued difference equations are controllable and observable in the sense of quaternion-valued and complex numbers, respectively. In addition, a direct method to solve the control function and duality is also given. Finally, we illustrate our theoretical results with some examples.

A Novel Error Criterion of Fundamental Matrix Based on Principal Component Analysis

Estimating the fundamental matrix (FM) using the known corresponding points is a key step for three-dimensional (3D) scene reconstruction, and its uncertainty directly affects camera calibration and point-cloud calculation. The symmetric epipolar distance is the most popular error criterion for estimating FM error, but it depends on the accuracy, number, and distribution of known corresponding points and is biased. This study mainly focuses on the error quantitative criterion of FM itself. First, the calculated FM process is reviewed with the known corresponding points. Matrix differential theory is then used to derive the covariance equation of FMs in detail. Subsequently, the principal component analysis method is followed to construct the scalar function as a novel error criterion to measure FM error. Finally, three experiments with different types of stereo images are performed to verify the rationality of the proposed method. Experiments found that the scalar function had approximately 90% correlation degree with the Manhattan norm, and greater than 80% with the epipolar geometric distance. Consequently, the proposed method is also appropriate for estimating FM error, in which the error ellipse or normal distribution curve is the reasonable error boundary of FM. When the error criterion value of this method falls into a normal distribution curve or an error ellipse, its corresponding FM is considered to have less error and be credible. Otherwise, it may be necessary to recalculate an FM to reconstruct 3D models.

Detecting “invisible” layers in layered composites

Lamb waves propagating in a layered structure consisting of layers with equal acoustic impedances, is studied by a variant of Cauchy sextic formalism and the exponential fundamental matrix method. It is shown that the dispersion portraits provide enough information to distinguish individual layers, even if their acoustic impedances are identical, making such layers indistinguishable by the pulse – echo method. It is revealed that the developed methodology is adequate for determining geometrical and physical properties of the individual layers with equal acoustic impedances.

Quaternion-valued exponential matrices and its fundamental properties

Quaternion differential equations (QDEs) are a new kind of differential equations which differ from ordinary differential equations. Our aim is to get the exponential matrices for the QDE which is useful for finding the solution of quaternion-valued differential equations, also, we know that linear algebra is very useful to calculate the exponential for a matrix but the solution of QDE is not a linear space. Due to the noncommutativity of the quaternion, the solution set of QDE is a right free module. For this, we must read some basic concepts on Quaternions such as eigenvalues, eigenvectors, Wronskian and the difference between quaternion and complex eigenvalues and eigenvectors; by using the right eigenvalue method for quaternions we developed a fundamental matrix which is useful to construct the exponential matrices which perform a great role in solving the QDEs.

A Mathematical Model Analysis for Estimating Stock Market Price Changes

In this paper, different methods for estimation of parameters of Weibull distribution were examined using Mean Square Error (MSE) as a criterion for selecting the best model. The Method of Moments exceeded other methods. In the same circumstance, the estimated results were logically extended to form a matrix that would help in predicting different commodity price processes by exploring the properties of fundamental matrix solution where we obtained predicted stock prices and asset returns for 12 months. Finally, from the fundamental matrix system a theorem was developed and proved to show different levels of changes as it affects stock market in terms of short-run and long-run respectively

A Taylor-Splitting Collocation approach and applications to linear and nonlinear engineering models

A novel matrix method based on the Taylor series called Taylor-Splitting Collocation Method is presented to solve linear and nonlinear ordinary differential equations. Unlike the previous approaches, the fundamental matrix equation is reformulated using interval splitting. The residual error estimation algorithm is presented. Convergence analysis is given in a general form. Four different mechanical models are analyzed:1.Forced oscillations of a linear spring-mass model2.Forced oscillations of a nonlinear spring-mass model3.Free oscillations of a cubic nonlinear spring-dashpot-mass model4.Forced oscillations of a damped nonlinear pendulum modelDisplacement-time and velocity-time dependencies are plotted for each model. Phase portraits of nonlinear models are presented. Appropriate absolute or residual error analyses are obtained for the proposed application models. The results of the new solution approach are compared with exact, numerical, and approximate solutions from previous works. Consistent results are found.

Abstract A025: Profiling tumor infiltrating immune cells for better understanding tumor status and better response to therapeutic strategy in soft tissue sarcomas

Abstract The tumor microenvironment is closely associated with tumor development and progression, which is made up of tumor cells, blood vessels, immune cells, fibroblast, and so on. Especially, immune cells within tumor microenvironments have been attracted in clinicians and researchers for unexpected therapeutic responses. Even in sarcoma, which is rare, and heterogenous leading to few options for cure, immune therapy has been tried and shown to be promising. However, the information of immune cells within soft tissue sarcomas were limited, which mostly have been obtained and characterized by omics-based analyses or tissue staining such as immunohistochemistry. Here, we established platform to profile tumor infiltrating immune cells by flow cytometry. About thirty sarcoma tissues were obtained, in which tumor infiltrating immune cells were profiled. The profiled data were confirmed by immunohistochemistry for correlation. It was shown that infiltrated leukocytes and CD3+ T cells were remarkably increased, whereas NKT and CD4+ T cells were decreased in tumor tissues compared to adjacent normal tissues. M1 macrophage was also increased in tumor tissues. High CD3+ T cell population was positively correlated with CD8+ T cells and PD1+ CD8+ T cell population. Especially, PD1+CD8+ T cells are usually expressed exhaustion-related conditions and low levels of cytotoxic immune cells. This study provides resources to understand immune cell population of soft tissue sarcoma, and will gain a fundamental matrix to study immunotherapy. These studies might show the possibility to find potential prognostic and immunotherapeutic targets. Citation Format: Eun-Young Lee, Hyun Guy Kang, June Hyuk Kim, Jong Woong Park, Seog Yun Park, Tak Yun, Hye Jin You. Profiling tumor infiltrating immune cells for better understanding tumor status and better response to therapeutic strategy in soft tissue sarcomas [abstract]. In: Proceedings of the AACR Special Conference: Sarcomas; 2022 May 9-12; Montreal, QC, Canada. Philadelphia (PA): AACR; Clin Cancer Res 2022;28(18_Suppl):Abstract nr A025.

Generalized Helical Hypersurfaces Having Time-like Axis in Minkowski Spacetime

In this paper, the generalized helical hypersurfaces x=x(u,v,w) with a time-like axis in Minkowski spacetime E14 are considered. The first and the second fundamental form matrices, the Gauss map, and the shape operator matrix of x are calculated. Moreover, the curvatures of the generalized helical hypersurface x are obtained by using the Cayley–Hamilton theorem. The umbilical conditions for the curvatures of x are given. Finally, the Laplace–Beltrami operator of the generalized helical hypersurface with a time-like axis is presented in E14.

Solution approaches for evaporation-driven density instabilities in a slab of saturated porous media

This work considers the gravitational instability of a saline boundary layer formed by an evaporation-induced flow through a fully saturated porous slab. Evaporation of saline waters can result in the formation of salt lakes as salt accumulates near the top surface. As gravitational instabilities can impede the accumulation of salt, establishing a relation between their occurrence and the value of physical parameters such as evaporation rate is crucial. One step toward determining when gravitational instabilities can arise is to compute the ground-state salinity. In this work, this ground state is analytically derived within the framework of Sturm–Liouville theory. Then, a linear stability analysis in conjunction with the quasi-steady state approach is employed to investigate the occurrence of instabilities. These instabilities can develop and grow over time depending on the Rayleigh number and the dimensionless height of the porous medium. To calculate the critical Rayleigh number, which is used to determine the stability of a particular system, the eigenvalues of the linear perturbation equations have to be computed. Here, a novel fundamental matrix method is proposed to solve this perturbation eigenvalue problem and shown to coincide with an established Chebyshev–Galerkin method in their shared range of applicability. Finally, a two-dimensional direct numerical simulation of the full equation system via the finite volume method is employed to validate the time of onset of convective instabilities predicted by the linear theory. Moreover, the fully nonlinear convection patterns are analyzed.

MODIFICATION OF THE PARAMETRIZATION METHOD FOR SOLVING A BOUNDARY VALUE PROBLEM FOR LOADED DIFFERENTIAL EPCAG

In this paper, modification of Dzhumabaev parameterization method is developed to a boundary value problem for systems of loaded differential equations with piecewise constant argument of generalized type (EPCAG). The method is based on reducing the considering problem to an equivalent multi-point boundary value problem for ordinary differential equations with parameters. An equivalent boundary value problem with parameters consists of the Cauchy problem for a system of ordinary differential equations with parameters, a two-point condition, a continuity condition, and additional conditions for a piecewise constant argument. The solution of the Cauchy problem for a system of ordinary differential equations with parameters is constructed using the fundamental matrix of the differential equation. A system of linear algebraic equations for the parameters is compiled using the values of the solution at the corresponding points and substituting them into the two-point condition, the continuity condition, and the conditions for the piecewise constant argument. A modification of Dzhumabaev parameterization method for solving the considering boundary value problem is proposed which is based on solving the constructed system and the 4th order Runge-Kutta method for solving the Cauchy problem on subintervals. The obtained results are verified by a numerical example. Numerical analysis showed high efficiency of the constructed modification of Dzhumabaev parameterization method.

Eigenvalue method for the solution of multi-compartment model of HIV-AIDS transmission

The aim of this study is to formulate a multi-compartment mathematical model regarding the transmission and dynamics of HIV-AIDS. The model is formulated on the basis of a system of linear, ordinary differential equations and admits two locally and globally stable equilib-ria. Primarily, the existence and uniqueness of solution of the model are demonstrated which is then obtained analytically using the fundamental matrix method and eigenvalue approach. The obtained solution serves as the pedestal for studying the dynamics and spread of HIV-AIDS in gen-eral. Nevertheless, as an endorsement to the obtained results the simu-lations are also carried out with model outcomes being contrasted to the exact data of the disease in India.

A mathematical model to study thermoregulation and heat-transfer processes in hypothermic neonates under variable physiological parameters

All neonates, whether term or preterm, are prone to hypothermia, which is a major health issue causing many health problems to infants and sometimes even death. Thus, such subjects are imperative to study to help researchers and biologists in neonatology, for developing certain methods, procedures and devices to prevent these abnormal temperature fluctuations to save the neonates from this health threat. To this purpose, a multi-node mathematical model is developed, to provide detailed insights and its applications to study the temperature profiles, thermoregulatory and heat-transfer mechanisms in hypothermic neonates. The model is constructed using the radial form of heat equation along with appropriate boundary and initial conditions. The model solution is obtained with the aid of the variational finite element method followed by the fundamental matrix method. The model outcomes obtained show the temperature fluctuations and tissue responses in hypothermic neonates. Finally, the model outcomes are compared with the published/experimental work to prove the feasibility and validity of the proposed work. Moreover, this work generalizes the several previously published works in the related field.

First passage in discrete-time absorbing Markov chains under stochastic resetting

First passage of stochastic processes under resetting has recently been an active research topic in the field of statistical physics. However, most of previous studies mainly focused on the systems with continuous time and space. In this paper, we study the effect of stochastic resetting on first passage properties of discrete-time absorbing Markov chains, described by a transition matrix Q between transient states and a transition matrix R from transient states to absorbing states. Using a renewal approach, we exactly derive the unconditional mean first passage time (MFPT) to any absorbing state, the splitting probabilities and the conditional MFPT to a specific absorbing state without hitting any other absorbing state. All the quantities can be expressed in terms of a deformed fundamental matrix and R, where I is the identity matrix, and γ is the resetting probability at each time step. We further show a sufficient condition under which the unconditional MPFT can be optimized by stochastic resetting. Finally, we apply our results to two concrete examples: symmetric random walks on one-dimensional lattices with absorbing boundaries and voter model on complete graphs.

A three‐phase laminated anisotropic elliptical inhomogeneity with internal uniform stress resultants

AbstractWe use the Stroh octet formalism for the coupled stretching and bending deformations of Kirchhoff laminated anisotropic thin plates together with the complex variable formulations of the decoupled stretching and bending deformations of Kirchhoff homogeneous isotropic thin plates to study the internal stress resultants inside a laminated anisotropic elliptical inhomogeneity bonded to a surrounding infinite laminated anisotropic matrix through a confocal interphase layer composed of a homogeneous isotropic plate when the matrix is subjected to uniform remote membrane stress resultants and bending moments. We prove that for given geometric and material parameters of the composite, the internal stress resultants inside the elliptical inhomogeneity are uniform when four specific conditions on the remote loading are satisfied. Furthermore, the internal uniform elastic field is determined in real‐form in terms of the two 8 × 8 fundamental elastic plate matrices for both the inhomogeneity and the matrix and the three 4 × 4 real matrices S, H, and L for the matrix.

An anisotropic elastic parabolic inhomogeneity under uniform heat flux

Using the extended Stroh sextic formalism for thermo-anisotropic elasticity, we study the problem of an anisotropic elastic parabolic inhomogeneity embedded in an infinite anisotropic elastic matrix with separate thermomechanical properties when the matrix is subjected to uniform remote heat flux. We prove that the induced temperature, stresses and strains inside the parabolic inhomogeneity all take the form of the sum of a constant plus linear distributions of the two in-plane coordinates. Furthermore, the internal thermoelastic field of displacements, stresses and strains inside the parabolic inhomogeneity is obtained in explicit real form in terms of the fundamental elasticity matrices and the thermal moduli for both the inhomogeneity and the matrix.

A Global Fundamental Matrix Estimation Method of Planar Motion Based on Inlier Updating.

A fundamental matrix estimation based on matching points is a critical problem in epipolar geometry. In this paper, a global fundamental matrix estimation method based on inlier updating is proposed. Firstly, the coplanar constraint was incorporated into the solution of the fundamental matrix to reduce the number of parameters to be solved. Subsequently, an inlier updating matrix was introduced according to the threshold of the epipolar geometry distance to eliminate the potential outliers and obtain a reliable initial value of the fundamental matrix. On this basis, we employed a four-point iterative method to estimate the fundamental matrix and make it satisfy the rank constraint at the same time. Finally, the epipolar geometry in binocular vision was extended to triple-view, and the fundamental matrix obtained in the previous step was globally optimized by minimizing the coordinate deviation between the intersection point and feature point in each group of images. The experiments show that the proposed fundamental matrix estimation method is robust to noise and outliers. In the attitude measurement, the maximum static error was 0.104° and dynamic measurement error was superior to 0.273°, which improved the reconstruction accuracy of feature points. Indoor images were further used to test the method, and the mean rotation angle error was 0.362°. The results demonstrate that the estimation method proposed in this paper has a good practical application prospect in multi-view 3D reconstruction and visual localization.

The Weyl matrix balls corresponding to the matricial truncated Hamburger moment problem

The main goal of the paper is to parametrize the Weyl matrix balls associated with an arbitrary matricial truncated Hamburger moment problem. For the special case of a non-degenerate matricial truncated Hamburger moment problem, the corresponding Weyl matrix balls were computed by I. V. Kovalishina [34] in the framework of V. P. Potapov's method of ‘Fundamental matrix inequalities’.

Mean hitting time formula for positive maps

In the classical theory of Markov chains, one may study the mean time to reach some chosen state, and it is well-known that in the irreducible, finite case, such quantity can be calculated in terms of the fundamental matrix of the walk, as stated by the mean hitting time formula. In this work, we present an analogous construction for the setting of irreducible, positive, trace preserving maps. The reasoning on positive maps generalizes recent results given for quantum Markov chains, a class of completely positive maps acting on graphs, presented by S. Gudder. The tools employed in this work are based on a proper choice of block matrices of operators, inspired in part by recent work on Schur functions for closed operators on Banach spaces, due to F. A. Grünbaum and one of the authors. The problem at hand is motivated by questions on quantum information theory, most particularly the study of quantum walks, and provides a basic context on which statistical aspects of quantum evolutions on finite graphs can be expressed in terms of the fundamental matrix, which turns out to be an useful generalized inverse associated with the dynamics. As a consequence of the wide generality of the mean hitting time formula found in this paper, we have obtained extensions of the classical version, either by assuming only the knowledge of the probabilistic distribution for the initial state, or by enlarging the arrival state to a subset of states.

Определение функций чувствительности вектора фазовых координат линейной динамической системы на основе интегро-степенного ряда по матрице вариаций параметров

The paper introduces a method for calculating sensitivity functions of the first and second orders of phase coordinates that do not require the integration of cumbersome chain-related systems of differential equations. The general and particular solutions of the vector differential equation of motion, written in the Cauchy form, are expressed in terms of the fundamental matrix. With the help of formal transformations, the vector of phase coordinates is represented as a convergent integro-power series with respect to the matrix that determines the variations of the elements of the matrix of system equation coefficients. In this work, the system is called the variation matrix. Then the relations obtained are transformed by special operations to an explicit form with respect to these variations up to and including the quadratic approximation. Within the framework of the matrix apparatus, there are similar expansions in terms of variations of the external influence and initial conditions. To exclude numerous "parasitic" operations of multiplication by zero in calculations, it is proposed to use special operations of matrix algebra. The original and inverse fundamental matrices are treated as multiplicative integrals, which ensures their simple calculation in time using recurrent formulas. The resulting apparatus is built entirely on matrix operations, which ensures simple machine implementation and versatility.