Matrix Algebra Techniques Research Articles

On the solution of MHD Jeffery–Hamel problem involving flow between two nonparallel plates with a blood flow application

AbstractThe Jeffery–Hamel flow phenomenon appears in a variety of real‐world applications involving the flow of two nonparallel plates. BY using a similarity transformation derived from the equation of continuity, partial differential equations determining flow characteristics are translated into nonlinear ordinary differential equations. The problem involves the flow of a specific type of fluid, namely, an incompressible and electrically conducting fluid, between two nonparallel plates. The flow is assumed to be steady, two‐dimensional, and subject to certain boundary conditions. Specifically, the plates are impermeable, and the fluid adheres to a no‐slip condition, resulting in zero fluid velocity at the plates' surfaces. Moreover, the problem incorporates the effects of magnetic fields and pressure fluctuations, making it highly applicable to scenarios, such as blood flow through arteries in the human body, which can be modeled as a special case of the magnetohydrodynamic (MHD) Jeffery–Hamel problem referred to as the (MHD) blood pressure equation. This work compares two numerical approaches for solving the MHDs Jeffery–Hamel problem: B‐spline and Bernstein polynomial collocation. The given approaches are used to discretize and transform the equation into a system of algebraic equations. Matrix algebra techniques are then used to solve the resultant system. A complete error analysis and convergence rates for different grid sizes are derived for both methods and are used to compare the accuracy and efficiency of the two approaches. Both approaches produce correct solutions, according to the numerical findings, although the Bernstein polynomial collocation method is more efficient and accurate than the B‐spline collocation.

The altered Hermite matrix: implications and ramifications

<p>Matrix theory is essential for addressing practical problems and executing computational tasks. Matrices related to Hermite polynomials are essential due to their applications in quantum mechanics, numerical analysis, probability, and signal processing. Their orthogonality, recurrence relations, and spectral properties make them a valuable tool for both theoretical research and practical applications. From a different perspective, we introduced a variant of the Hermite matrix that incorporates triple factorials and demonstrated that this matrix satisfies various properties. By utilizing effective matrix algebra techniques, various algebraic properties of this matrix have been determined, including the product formula, inverse matrix and eigenvalues. Additionally, we extended this matrix to a more generalized form and derived several identities.</p>

Hypersurfaces of revolution family supplying $ \Delta \mathfrak{r} = \mathcal{A}\mathfrak{r} $ in pseudo-Euclidean space $ \mathbb{E}_{3}^{7} $

<abstract><p>In this study, we introduce a family of hypersurfaces of revolution characterized by six parameters in the seven-dimensional pseudo-Euclidean space $ {\mathbb{E}}_{3}^{7} $. These hypersurfaces exhibit intriguing geometric properties, and our aim is to analyze them in detail. To begin, we calculate the matrices corresponding to the fundamental form, Gauss map, and shape operator associated with this hypersurface family. These matrices provide essential information about the local geometry of the hypersurfaces, including their curvatures and tangent spaces. Using the Cayley-Hamilton theorem, we employ matrix algebra techniques to determine the curvatures of the hypersurfaces. This theorem allows us to express the characteristic polynomial of a matrix in terms of the matrix itself, enabling us to compute the curvatures effectively. In addition, we establish equations that describe the interrelation between the mean curvature and the Gauss-Kronecker curvature of the hypersurface family. These equations provide insights into the geometric behavior of the surfaces and offer a deeper understanding of their intrinsic properties. Furthermore, we investigate the relationship between the Laplace-Beltrami operator, a differential operator that characterizes the geometry of the hypersurfaces, and a specific $ 7\times 7 $ matrix denoted as $ \mathcal{A} $. By studying this relation, we gain further insights into the geometric structure and differential properties of the hypersurface family. Overall, our study contributes to the understanding of hypersurfaces of revolution in $ {\mathbb{ E}}_{3}^{7} $, offering mathematical insights and establishing connections between various geometric quantities and operators associated with this family.</p></abstract>

Asset Allocation with Liquidity-Adjusted Market Risk Modeling: Empirical Relevance to Multiple-Asset Portfolios

In this paper, the author demonstrate a practical approach for measurement, management and control of market risk exposure for financial trading portfolios. This approach is based on the renowned concept of Liquidity-Adjusted Value at Risk (L-VaR) along with the creation of a software tool utilizing matrix-algebra technique under the notion of different correlation factors and liquidation horizons.

Analyzing the stability of gene expression using a simple reaction-diffusion model in an early Drosophila embryo

In all complex organisms, the precise levels and timing of gene expression controls vital biological processes. In higher eukaryotes, including the fruit fly Drosophila melanogaster, the complex molecular control of transcription (the synthesis of RNA from DNA) and translation (the synthesis of proteins from RNA) events driving this gene expression are not fully understood. In particular, for Drosophila melanogaster, there is a plethora of experimental data, including quantitative measurements of both RNA and protein concentrations, but the precise mechanisms that control the dynamics of gene expression during early development and the processes which lead to steady-state levels of certain proteins remain elusive.This study analyzes a current mathematical modeling approach in an attempt to better understand the long-term behavior of gene regulation. The model is a modified reaction-diffusion equation which has been previously employed in predicting gene expression levels and studying the relative contributions of transcription and translation events to protein abundance [10,11,24]. Here, we use Matrix Algebra and Analysis techniques to study the stability of the gene expression system and analyze equilibria, using very general assumptions regarding the parameter values incorporated into the model. We prove that, given realistic biological parameter values, the system will result in a unique, stable equilibrium solution. Additionally, we give an example of this long-term behavior using the model alongside actual experimental data obtained from Drosophila embryos.

Block invariance and reversibility of one dimensional linear cellular automata

Consider a one-dimensional, binary cellular automaton f (the CA rule), where its n nodes are updated according to a deterministic block update (blocks that group all the nodes and such that its order is given by the order of the blocks from left to right and nodes inside a block are updated synchronously). A CA rule is block invariant over a family F of block updates if its set of periodic points does not change, whatever the block update of F is considered. In this work, we study the block invariance of linear CA rules by means of the property of reversibility of the automaton because such a property implies that every configuration has a unique predecessor, so, it is periodic. Specifically, we extend the study of reversibility done for the Wolfram elementary CA rules 90 and 150 as well as, we analyze the reversibility of linear rules with neighbourhood radius 2 by using matrix algebra techniques.

COMPARISON OF VSM, GVSM, AND LSI IN INFORMATION RETRIEVAL FOR INDONESIAN TEXT

Vector space model (VSM) is an Information Retrieval (IR) system model that represents query and documents as n-dimension vector. GVSM is an expansion from VSM that represents the documents base on similarity value between query and minterm vector space of documents collection. Minterm vector is defined by the term in query. Therefore, in retrieving a document can be done base on word meaning inside the query. On the contrary, a document can consist the same information semantically. LSI is a method implemented in IR system to retrieve document base on overall meaning of users’ query input from a document, not based on each word translation. LSI uses a matrix algebra technique namely Singular Value Decomposition (SVD). This study discusses the performance of VSM, GVSM and LSI that are implemented on IR to retrieve Indonesian sentences document of .pdf, .doc and .docx extension type files, by using Nazief and Adriani stemming algorithm. Each method implemented either by thread or no-thread. Thread is implemented in preprocessing process in reading each document from document collection and stemming process either for query or documents. The quality of information retrieval performance is evaluated based-on time response, values of recall, precision, and F-measure were measured. The results show that for each method, the fastest execution time is .docx extension type file followed by .doc and .pdf. For the same document collection, the results show that time response for LSI is more faster, followed by GVSM then VSM. The average of recall value for VSM, GVSM and LSI are 82.86 %, 89.68 % and 84.93 % respectively. The average of precision value for VSM, GVSM and LSI are 64.08 %, 67.51 % and 62.08 % respectively. The average of F-measure value for VSM, GVSM and LSI are 71.95 %, 76.63 % and 71.02 % respectively. Implementation of multithread for preprocessing for VSM, GVSM, and LSI can increase average time response required is about 30.422%, 26.282%, and 31.821% respectively.

Scenario optimization technique for the assessment of downside-risk and investable portfolios in post-financial crisis

The aim of this paper is to develop an optimization technique for the assessment of downside-risk limits and investable financial portfolios under crisis-driven outlooks subject to applying meaningful financial and operational constraints. The simulation and testing methods are based on the renowned concept of liquidity-adjusted value-at-risk (LVaR) along with the development of an optimization risk-algorithm utilizing matrix–algebra technique. With the purpose of demonstrating the effectiveness of LVaR and stress-testing techniques, real-world quantitative analysis of structured equity portfolios are depicted for the Gulf Cooperation Council (GCC) financial markets. To this end, several structural simulations studies are accomplished with the goal of establishing realistic financial modeling algorithm for the calculation of downside-risk parameters and to empirically assess portfolio managers' optimal and investable portfolios. The developed methodology and risk valuation algorithms can aid in advancing risk assessment and portfolio management practices in emerging markets, particularly in the wake of the most recent credit crunch and the subsequent financial turmoil.

Classification of KdV Vessels with Constant Parameters and Two Dimensional Outer Space

In this article we classify vessels producing solutions of some completely integrable partial differential equations (PDEs), presenting a unified approach for them. The classification includes such important examples as Korteweg-de Vries (KdV) and evolutionary non linear Schrodingier (ENLS) equations. In fact, employing basic matrix algebra techniques it is shown that there are exactly two canonical forms of such vessels, so that each canonical form generalize either KdV or ENLS equations. Particularly, Dirac canonical systems, whose evolution was recently inserted into the vessel theory, are shown to be equivalent to the ENLS equation in the sense of vessels. This work is important as a first step to a classification of completely integrable PDEs, which are solvable by the theory of vessels. We note that a recent paper of the author, published in Journal of Mathematical Physics, showed that initial value problem with analytic initial potential for the KdV equation has at least a “narrowing” in time solution. The presented classification inherits this idea and a similar theorem can be easily proved for the presented PDEs. Finally, the results of the work serve as a basis for the investigation of the following problems: (1) hierarchy of the generalized KdV, ENLS equations (by generalizing the vessel equations), (2) new completely integrable PDEs (by changing the dimension of the outer space), (3) addressing the question of integrability of a given arbitrary PDE (the future classification will create a list of solvable by vessels equations, which may eventually include many existing classes of PDEs).

A kinematic analysis of the gmf a-510 robot: An introduction and application of groebner basis theory

In this paper, we examine the application of a Groebner Basis to solve an inverse problem in mobility and control. Traditionally, such inverse problems are solved using matrix algebra techniques. This paper will introduce and solve an inverse problem using both matrix algebra and a Groebner Basis, in order to demonstrate the advantages of using a Groebner basis to: 1) classify the solutions of mobility, 2) find all solutions more efficiently, 3) and find all solutions with a fewer number of operations.

Improved Quantitative Modeling of Ligand-Activated Macromolecular Receptors using Conditional Binding

Experimental investigations of ligand-activated receptors are often based on binding measurements that detect the total fraction of occupied binding sites at various ligand concentrations. We use calmodulin, an important calcium sensor protein, as a test system to document the uncertainties in model parameters estimated by fitting various models to total binding data. Using nonlinear least-squares methods, we obtain excellent fits (<1% RMS error) to synthetic total binding data with the same characteristics as the published binding data using parameter sets with binding affinities varying by over four orders of magnitude for each site. This result identifies a significant obstacle blocking progress toward the goal of developing accurate, quantitative models of receptor activation. The use of noiseless data in our analysis suggests that the large uncertainties in the estimated parameters are not a problem of data quality, but rather reflect an intrinsic limitation of total binding data. Using analytical matrix algebra techniques and numerical simulations, we discover a fundamental relationship between the mathematical structure of the equations describing various types of binding data and the number and type of parameters that may be determined accurately from regression analysis of that data. Ideas based on Boolean logical principles are used to design a new type of binding experiment that significantly improves upon total binding data in its power to constrain physically realistic models of receptor activation. These experiments, which we call conditional binding, report on the simultaneous occupancies of two different receptor binding sites. Our approach is general and the conclusions are applicable to the many macromolecular systems that are activated or modulated by ligand binding.

Study on Multi-Exponential Inversion Method for NMR Relaxation Signals with Tikhonov Segularization

The analysis of NMR data in terms of inversion of relaxation distribution is hampered by the ill-posed nature of the required solution about the Fredholm integral equation. Naive approaches such as multi-exponential fitting or standard least-squares algorithms are numerically unstable and often failed. This paper updates the application of Tikhonov regularization to stabilize this numerical inversion problem and demonstrates the method for automatically choosing the optimal value of the regularization parameter. The approach is computationally efficient and easy to implement using standard matrix algebra techniques, which is based on mathematical ware MATLAB. Example analyses arepresented using both synthetic data and experimental results of NMR studies on the liquid samples like as oils and yoghurt.

Habitat suitability model for red deer (Cervus elaphus Linnaeus, 1758): spatial multi-criteria analysis with GIS application

Monitoring population trends is essential in wildlife management. In fact, to identify those environmental conditions linked to habitat use and selection is a key task in any conservation plan. Our aim in this work was predict red deer (Cervus elaphus Linnaeus, 1758) colonization in the Lombada National Hunting Area (LNHA), one of the largest wild populations in Portugal. The sharply increase in density and range of red deer over the last decades justify the careful monitoring of this population. A multi-criteria model using the Analytical Hierarchic Process (AHP) and GIS weighted spatial analysis with matrix algebra techniques were applied in this approach. This method was developed in three distinct phases: i) setting the objectives, ii) analysing pairs of factors through the application of a comparison matrix and iii) its synthesis with the weight assignment that followed a predetermined numeric range according to the ecological requirements of the study species. The variables used in the red deer habitat suitability model included the hydrographic network, asphalted roads, relief aspects and land use. From October 2010 to March 2011, a total of 129 observations along line transects were recorded and used to validate the final model. By using the AUC method it was obtained a success rate of 72%. Our results suggest that this method would be applied on a larger scale being suitable to predict red deer expansion. To our knowledge, this is the first study performed in Portugal using this methodological approach to assess the red deer-habitat relationships.

Habitat suitability model for red deer (Cervus elaphus Linnaeus, 1758): spatial multi-criteria analysis with GIS application

Monitoring population trends is essential in wildlife management. In fact, to identify those environmental conditions linked to habitat use and selection is a key task in any conservation plan. Our aim in this work was predict red deer (Cervus elaphus Linnaeus, 1758) colonization in the Lombada National Hunting Area (LNHA), one of the largest wild populations in Portugal. The sharply increase in density and range of red deer over the last decades justify the careful monitoring of this population. A multi-criteria model using the Analytical Hierarchic Process (AHP) and GIS weighted spatial analysis with matrix algebra techniques were applied in this approach. This method was developed in three distinct phases: i) setting the objectives, ii) analysing pairs of factors through the application of a comparison matrix and iii) its synthesis with the weight assignment that followed a predetermined numeric range according to the ecological requirements of the study species. The variables used in the red deer habitat suitability model included the hydrographic network, asphalted roads, relief aspects and land use. From October 2010 to March 2011, a total of 129 observations along line transects were recorded and used to validate the final model. By using the AUC method it was obtained a success rate of 72%. Our results suggest that this method would be applied on a larger scale being suitable to predict red deer expansion. To our knowledge, this is the first study performed in Portugal using this methodological approach to assess the red deer-habitat relationships.

Risk Management in Trading and Investment Portfolios

The aim of this article is to bridge the gap in equity trading risk management literatures and particularly from the perspective of emerging and illiquid markets, such as in the context of the Gulf Cooperation Council (GCC) financial markets. In this article, we demonstrate a practical approach for the measurement and control of market risk exposure for financial trading portfolios that contain several illiquid equity securities during the unwinding (close-out) period. This approach is based on the renowned concept of Value-at-Risk (VaR) along with the development of an optimisation risk algorithm utilising matrix-algebra technique. Our thorough asset market risk modelling-algorithm can simultaneously handle VaR analysis under normal and severe market conditions, besides it takes into account the effects of illiquidity and short-sales of traded equity securities. In order to illustrate the proper use of VaR and stress-testing methods, real-world structured modelling techniques of trading risk management are presented for the GCC financial markets. To this end, comprehensive simulation case studies are accomplished with the objective of constructing a realistic framework for trading risk measurement and control in addition to the instigation of a risk optimisation algorithm-process for the computation of maximum authorised VaR risk-budgeting limits. JEL Classification: C10, C13, G20, G28

A Framework for the Evaluation of Inclinometer Data in the Measurement of Structures

This paper presents a new algebraic framework for modeling structures and computing the reconstruction of curves from local derivatives. This is the fundamental problem behind the use of inclinometers to measure the deformation of structures. A solid theoretical background, with derivations, is provided for the solution of a real and practical measurement problem. The reconstruction problem is solved using matrix algebraic techniques applied to variational calculus. Tikhonov, spectral, and a new regularization technique based on truncated constrained basis functions are implemented. It is proved that the solution of differential equations via basis functions corresponds to computing the spectra of the bending modes with respect to the basis functions used. This insight yields a method of determining the optimal set of basis functions based on spectral compactness. A completely new approach to computing a linear differential operator which yields a global estimator is presented. The method is also extended to yield a regularizing linear differential operator. The methods are verified, and the claims made substantiated by extensive numerical simulations including Monte Carlo simulations to verify the susceptibility of the solution to noise. The methods are also applied to a real measurement problem of determining the subsidence of rail track during construction in the vicinity of the track. Simultaneous reconstruction and temporal filtering is implemented.

Evaluation of windowed ESPRIT virtual instrument for estimating Power Quality Indices

The use of subspace and Matrix Algebra (MA) methods for estimating Power Quality Indices (PQIs) as defined by IEEE Std 1459-2010 is relatively new. For this reason there is room for further research about these methods and their efficiency applied to estimating PQIs. This document presents the design of a Virtual Instrument (VI) based on modern subspace, MA and Least Square (LS) techniques with a Kalman Filter (KF) for estimating PQIs. Further we claim that the efficiency of the subspace method is comparable to the Fourier based methods in typical power system analysis. Simulation and experiments are conducted to further compare the performance of this VI with other VI's which do not use the KF. We also compare this VI with VI's that use the classical Fast Fourier Transform (FFT) based method of estimation.

Quantum convex support

Convex support, the mean values of a set of random variables, is central in information theory and statistics. Equally central in quantum information theory are mean values of a set of observables in a finite-dimensional C ∗-algebra A , which we call (quantum) convex support. The convex support can be viewed as a projection of the state space of A and it is a projection of a spectrahedron. Spectrahedra are increasingly investigated at least since the 1990s boom in semi-definite programming. We recall the geometry of the positive semi-definite cone and of the state space. We write a convex duality for general self-dual convex cones. This restricts to projections of state spaces and connects them to results on spectrahedra. Our main result is an analysis of the face lattice of convex support by mapping this lattice to a lattice of orthogonal projections, using natural isomorphisms. The result encodes the face lattice of the convex support into a set of projections in A and enables the integration of convex geometry with matrix calculus or algebraic techniques.

On the inversion of diffusion NMR data: Tikhonov regularization and optimal choice of the regularization parameter

The analysis of diffusion NMR data in terms of distributions of diffusion coefficients is hampered by the ill-posed nature of the required inverse Laplace transformation. Naïve approaches such as multiexponential fitting or standard least-squares algorithms are numerically unstable and often fail. This paper updates the CONTIN approach of the application of Tikhonov regularization to stabilise this numerical inversion problem and demonstrates two methods for automatically choosing the optimal value of the regularization parameter. These approaches are computationally efficient and easy to implement using standard matrix algebra techniques. Example analyses are presenting using both synthetic data and experimental results of diffusion NMR studies on the azo-dye sunset yellow and some polymer molecular weight reference standards.

A boundedness-preserving finite-difference scheme for a damped nonlinear wave equation

In this work, we present a non-standard finite-difference scheme to approximate solutions of a damped, hyperbolic partial differential equation with nonlinear reaction which follows a generalized logistic form. Our mathematical model is a hyperbolic generalization of the classical Fisher–Kolmogorov–Petrovsky–Piscounov equation from population dynamics, where the damping term has a constant coefficient; both zero Dirichlet and Neumann boundary conditions are considered. Under certain conditions of the model and the computational parameters, the method is able to preserve the positivity and the boundedness of the solutions of the mathematical model, as evidenced both theoretically (via techniques of matrix algebra which provide sufficient conditions on the positivity and the boundedness of the technique) and computationally (via numerical simulations). The validity of the method is tested against known exact solutions of the parabolic and the hyperbolic regimes, obtaining an excellent agreement between the analytical and numerical results.