Matrix Form Research Articles

Application of matrices in transport

The action of linear operators can be represented in a matrix, so it is most convenient to write systems of equations with a large number of unknowns in matrix form. In the following text, the basic operations with matrices will be listed, the existing computer tools in operations with matrices will be pointed out, and finally some practical problems from transportation that can be solved with matrices will be given. The transport chain can be formed by one or more types, with different cargo handling technologies, types of cargo units, types of carriers, all of which affect transport costs and delivery time. On the basis of techno-economic analysis, the best variants of transport chains are generated, which should be known by senders and receivers. The number of combinations can be important, and matrices can be used to determine the most favorable variants. The basic problems in the optimization of the stacking of piece loads (goods) on carriers are the appearance and mass at the places of initial and final operations, which in most cases are of stochastic size, and the distribution of load masses in a static and dynamic sense.

Unitary canonical forms over Clifford algebras, and an observed unification of some real-matrix decompositions

We show that the spectral theorem – which we understand to be a statement that every self-adjoint matrix admits a certain type of canonical form under unitary similarity – admits analogues over other ∗ -algebras distinct from the complex numbers. If these ∗ -algebras contain nilpotents, then it is shown that there is a consistent way in which many classic matrix decompositions – such as the Singular Value Decomposition, the Takagi decomposition, the skew-Takagi decomposition, and the Jordan decomposition, among others – are immediate consequences of these. If producing the relevant canonical form of a self-adjoint matrix were a subroutine in some programming language, then the corresponding classic matrix decomposition would be a 1-line invocation with no additional steps. We also suggest that by employing operator overloading in a programming language, a numerical algorithm for computing a unitary diagonalization of a complex self-adjoint matrix would generalize immediately to solving problems like SVD or Takagi. While algebras without nilpotents (like the quaternions) allow for similar unifying behaviour, the classic matrix decompositions which they unify are never obtained as easily. In the process of doing this, we develop some spectral theory over Clifford algebras of the form C l p , q , 0 ( R ) and C l p , q , 1 ( R ) where the former is admittedly quite easy. We propose a broad conjecture about spectral theorems.

Алгоритм оценивания публикационной активности преподавателей вузов, направленный на улучшение научно-педагогической деятельности

Purpose: To create a more effective algorithm for evaluating the publication activity of lecturers in comparison with the existing algorithms for the formation of the Hirsch index and ghp-index. Methods: The algorithm includes a mathematical description of all published works of the lecturer and their citations. The results of the scientific activity of the lecturer are reflected using the matrix form of recording statistical source data, consisting of citations for each published work of the author. The sequence of citations of publications in the matrix is determined by the non-increasing nature of citations in subsequent publications compared to the previous ones. The constructed matrix of the distribution of citations S by published works is rebuilt into a block matrix, which includes the following block matrices: H is the base matrix defining the Hirsch index; G is the matrix of significant publications located above the Hirsch matrix; P is the matrix of less-cited works of the lecturer; O is the zero matrix. The formed matrices have allowed the introduction of the following indices using Euclidean norms: h — Hirsch index, g — index of significant publications and p — index of intensive work. In turn, these indices have allowed to determine the following as Euclidean norms: gh — the index of basic significant publications, hp — the index of intensive work of a lecturer and a comprehensive ghp-index that takes into account all the published works of a lecturer and all their citations. The ghp-index has been improved by introducing weighting coefficients for its constituent indices. Results: The ratings of a team of 20 lecturers has been formed using a sample from the RSCI with the help of the Hirsch index and the A-ghp index. It is shown that the proposed algorithm is more effective in comparison with the Hirsch index and other introduced indexes. Practical significance: The conducted research has allowed to substantiate the possibility of using an improved comprehensive index to distribute the incentives of a team of lecturers in a differentiated and fair manner according to their rating. The algorithm aims to improve the scientific and methodological activities of lecturers by providing fair and accurate ranking of the members of their collectives.

Love stories in a differential equations classroom

We believe that developing cultural competencies can help students learn mathematics and conversely that learning mathematical content can help students learn about themselves and others. Using frameworks introduced by Rendón (2009) and Gutiérrez (2018), we present a five-part bundle of activities for undergraduate differential equations course instructors, including one pre-activity reflection assignment, three modeling activities, and a final project. The Pre-Activity Assignment engages students to draw on their own personal and/or cultural experiences with the concept of love. Three activities focus on developing revision skills in mathematical modeling and practicing methods of solving systems of ordinary differential equations (ODEs). In these activities, students would collaborate to construct a relationship model consisting of a system of first-order linear ODEs and solve different model iterations with the characteristic polynomial, matrix form, or Laplace transform method. The final project connects the reflections in Pre-Activity Assignment with the skills developed in the three activities by inviting students to create a relationship scenario, model, revise, solve it, and present the conclusions. By engaging in this set of assignments, students connect personal and cultural experiences with the concept of love and perceive themselves and their peers in the curriculum, fostering a sense of belonging and relevance.

A new hybrid method combining search and direct based construction ideas to generate all 4 × 4 involutory maximum distance separable (MDS) matrices over binary field extensions.

This article presents a new hybrid method (combining search based methods and direct construction methods) to generate all involutory maximum distance separable (MDS) matrices over . The proposed method reduces the search space complexity at the level of , where n represents the number of all invertible matrices over to be searched for. Hence, this enables us to generate all involutory MDS matrices over and . After applying global optimization technique that supports higher Exclusive-OR (XOR) gates (e.g., XOR3, XOR4) to the generated matrices, to the best of our knowledge, we generate the lightest involutory/non-involutory MDS matrices known over , and in terms of XOR count. In this context, we present new involutory MDS matrices over , and , which can be implemented by 13 XOR operations with depth 5, 25 XOR operations with depth 5 and 42 XOR operations with depth 4, respectively. Finally, we denote a new property of Hadamard matrix, i.e., (involutory and MDS) Hadamard matrix form is, in fact, a representative matrix form that can be used to generate a small subset of all involutory MDS matrices, where k > 1. For k = 1, Hadamard matrix form can be used to generate all involutory MDS matrices.

A stochastic alternative technique for Compton Maximum Likelihood Expectation-Maximization (MLEM) reconstruction

Even though iterative methods and particularly Maximum Likelihood Expectation-Maximization (MLEM) algorithms have been established in reconstruction with Compton data, their detailed design with respect to physical rules and processes dominate their plain implementation in the form of a system matrix. A new elementary but efficient alternative for the well-known system matrix with respect to Compton Camera image reconstruction is presented in this work. For each detected event there is a weighting factor inserted as an accumulated probability which carries all the necessary information. This probability which involves only the Compton scattering angle of the incident photon corresponds to a map that correlates all events within all possible source origins. Based on maximizing likelihood principles, the proposed model weights in a stochastic way the difference of the scatterer-to-source angle θ0, as it is determined by the deposited energy on the absorber, and any other potential scattering angle θJ, specified by the position coordinates on the reconstruction matrix. Obtained image spatial resolution, angular distortions and response to focal length determination are a few of the studied cases for the algorithms’ evaluation via simulations in GEANT4/GATE with a set of radioactive sources and phantoms with in- and out-of-plane arrangement.

Theta method applied to two-energy groups point kinetics equations

The point kinetics equations with two-energy groups and multi-group of delayed neutrons are a system of stiff linear/nonlinear ordinary differential equations strongly coupled, which is used to simulate the dynamics system of a homogeneous nuclear reactor by determining the population of neutrons and precursor groups. The matrix form of the mathematical model is solved by the theta method where the matrix inverse is calculated analytically. Numerical results are obtained through iterative schemes to approximate analytical solutions. The stiffness condition of the system of equations can cause instability in the solutions of explicit iterative schemes commonly employed so far. In this work, an implicit iterative scheme is used to approximate numerical solutions to the homogeneous nuclear reactor with one and six groups of delayed neutrons. The results are discussed and compared on light of literature reports. Unprecedented comparisons with experimental data from an AGN-201 nuclear reactor are also discussed. For the case studies, the theta method exhibits accurate and computationally efficient results.

A novel complex variable solution for the stress and displacement fields around a shallow non-circular tunnel

A novel complex variable solution is derived in this paper for the stress and displacement fields around a shallow non-circular tunnel. The proposed method allows explicit computations in matrix form and has advantages of considerable efficiency and high accuracy. Firstly, the stress and displacement boundary conditions for the shallow non-circular tunnel are obtained using the generalized complex variable theory and conformal mapping technique. Then, the fast Fourier transform algorithm is employed to effectively perform the Fourier series fitting for these boundary conditions, and matrix equations for the potential functions in the frequency domain are established. Later, a matrix solution for the potential functions is formulated to determine the stress and displacement fields around the shallow non-circular tunnel. Finally, select numerical analyses are conducted to verify the proposed method, discuss the key parameters in the algorithm, and reveal the effect of the acceleration strategy using FFT as well as the effect of non-circular geometries on the stress and displacement fields.

Constrained read-once refutations in UTVPI constraint systems: A parallel perspective

AbstractIn this paper, we analyze two types of refutations for Unit Two Variable Per Inequality (UTVPI) constraints. A UTVPI constraint is a linear inequality of the form: $a_{i}\cdot x_{i}+a_{j} \cdot x_{j} \le b_{k}$ , where $a_{i},a_{j}\in \{0,1,-1\}$ and $b_{k} \in \mathbb{Z}$ . A conjunction of such constraints is called a UTVPI constraint system (UCS) and can be represented in matrix form as: ${\bf A \cdot x \le b}$ . UTVPI constraints are used in many domains including operations research and program verification. We focus on two variants of read-once refutation (ROR). An ROR is a refutation in which each constraint is used at most once. A literal-once refutation (LOR), a more restrictive form of ROR, is a refutation in which each literal ( $x_i$ or $-x_i$ ) is used at most once. First, we examine the constraint-required read-once refutation (CROR) problem and the constraint-required literal-once refutation (CLOR) problem. In both of these problems, we are given a set of constraints that must be used in the refutation. RORs and LORs are incomplete since not every system of linear constraints is guaranteed to have such a refutation. This is still true even when we restrict ourselves to UCSs. In this paper, we provide NC reductions between the CROR and CLOR problems in UCSs and the minimum weight perfect matching problem. The reductions used in this paper assume a CREW PRAM model of parallel computation. As a result, the reductions establish that, from the perspective of parallel algorithms, the CROR and CLOR problems in UCSs are equivalent to matching. In particular, if an NC algorithm exists for either of these problems, then there is an NC algorithm for matching.

Dynamical behavior of a stochastic dengue model with Ornstein–Uhlenbeck process

We develop and study a stochastic dengue model with Ornstein–Uhlenbeck process, in which we assume that the transmission coefficients between vector and human satisfy the Ornstein–Uhlenbeck process. We first show that the stochastic system has a unique global solution with any initial value. Then we use a novel Lyapunov function method to establish sufficient criteria for the existence of a stationary distribution of the system, which indicates the persistence of the disease. In particular, under some mild conditions which are applied to ensure the local asymptotic stability of the endemic equilibrium of the deterministic system, we obtain the specific form of covariance matrix in the probability density around the quasi-positive equilibrium of the stochastic system. In addition, we also establish sufficient criteria for wiping out of the disease. Finally, several numerical simulations are performed to illustrate our theoretical conclusions.

Number of Solutions for Some Special Logical Analysis Problems of Integer Data

In the class of discrete enumeration problems, an important place belongs to the problems of searching for frequently and infrequently occurring elements in integer data. Questions on the effectiveness of such a search are directly related to the study of the metric (quantitative) properties of sets of frequent and infrequent elements. It is assumed that the initial data are presented in the form of an integer matrix, whose rows are descriptions of the studied objects in the given system of the numerical characteristics of these objects, called attributes. The case is considered when each attribute takes values from the set {0,1,...k-1}, k2. Asymptotic estimates for the typical number of special, frequent fragments of object descriptions, called correct fragments, and estimates for the typical length of such a fragment are given. We also present new results concerning the study of the metric properties of the minimal infrequent fragments of descriptions of objects.

The active control study of a damped two-level smart spring system

The vibration reduction performance of a damped two-level smart spring system is studied. The governing equations of the smart spring system are reduced to a first-order ordinary differential equation of matrix form. An active control algorithm employing the relative velocity relationships between the primary and auxiliary systems to prevent the movement of the controlled object is applied in the simulation. Different excitation types are considered to fully examine the validity of the control law. Responses of the controlled object under the control law are obtained and compared to the case of constant friction and the case of no control. The study indicates that the active control algorithm shows a good vibration reduction efficiency and a wide vibration reduction bandwidth.

Development of a spatial orientation model for the actuator of a mechatronic system

Purpose. Development of a control system for a mechatronic system, taking into account the geometric solution of the inverse kinematics problem. The methods. The research was based on the principles of theoretical and applied mechanics, as well as the Denavit-Hartenberg transformation model. The formation of graphical computational models and kinematic diagrams simplified the development of the mathematical model of the dynamic system without considering the forces acting on the mechanism links. The possibility of using matrix transformations to simplify the search for generalized coordinates and transition to the local disposition of each link was explored for further integration of the obtained results into automation and control systems. Findings. The obtained results, determining the trajectory of the gripper using inverse kinematics models, allowed the development of algorithms for determining the position of mechanism links with the possibility of software implementation of the control system. Identifying patterns for determining accurate position coordinates by the matrix method allows the application of open-source software for real-time position calculations. The originality. The use of modern technologies for visual assessment of the external environment and coordinating control impulses of the executive body drive was first achieved based on the mathematical model of the inverse kinematics of a multi-link mechanism. This automated the determination of local coordinates for each link within its degrees of freedom and algorithmized this process. Combining the mathematical models of mechanism kinematics and the matrix form of coordinate search allows investigating the influence of the displacement of the i-th link of the mechatronic technical complex on the overall system, considering the acting force systems and the specified spatial orientation of both the executive body and intermediate links. Practical implementation. The implementation of obtained kinematic models using the matrix method enables the software realization of algorithms for searching the coordinates of the i-th mechanism link, automating the control process with the task of final positions, determining the level of integral error during the displacement of the initial link, and providing the possibility of programming autonomous mechatronic systems with open-source code. As a result, the introduction of semi or fully autonomous technical complexes will automate technological processes in various industries.

Matrix representations of the inverse problem in the graph model for conflict resolution with fuzzy preference

The inverse analysis of the graph model for conflict resolution (GMCR) determines all the preference relationships required by each decision-maker (DM) according to a given stability definition to ensure that the given state is stable. In the paper, a matrix representation approach for the inverse graph model with fuzzy preference relations is proposed to obtain all the preference relations for each DM making the stability result based on the fuzzy stability definition, including fuzzy Nash, fuzzy general metarationality, fuzzy symmetric metarationality, and fuzzy sequential stability definition. For the two-DM graph model, the fuzzy preference relation over states, unilateral movements, and fuzzy unilateral improvements are refreshed by using the matrix forms. Furthermore, for the multiple DMs model, the joint movements and joint improvement for a collection including at least two DM are represented by using matrix representation. Finally, the matrix relationship for specifying the required preference relation in the framework of inverse graph model with fuzzy preferences is derived. To illustrate the usefulness of the matrix representation of inverse graph model with fuzzy preference relations, this paper demonstrates it using a real-life conflict of the Elmira groundwater contamination conflict.

A norm stability condition of neutral-type Cohen-Grossberg neural networks with multiple time delays

By constructing an appropriate Lyapunov functional, this paper obtains a novel delay-independent stability criterion of neutral-type Cohen-Grossberg neural networks possessing multiple time delays. Although this type of system cannot be represented as vector–matrix form due to the presence of multiple delays, our stability conclusion is fully defined by the infinite norm of parametric matrices and the network parameters first time. Due to the feasibility and simplicity of method proposed, our stability conclusion reducing the computational complexity while also reducing the conservatism compared with several onetime literature. Two concrete neural network models are applied to confirm the effectiveness and superiority of our conclusion.

Formation of germanene with free-standing lattice constant

Two-dimensional sheets of ultra-thin germanium with the lattice parameter of free-standing germanene, i.e., 0.41 nm, are formed by atomic segregation epitaxy on face-centered-cubic Ag0.9Al0.1(111) thin films grown on a Ge(111) template. Low-energy electron diffraction clearly shows a square-like (√13 × √13)R ± 46° superstructure, expressed as (4114) in Matrix form, whose 92° angle between the two vectors is clearly observed, as well as incommensurate (1.41 × 1.41)R ± 27° spots. Scanning tunneling microscopy (STM) images show the square-like superstructure and a defective honeycomb-like arrangement with this characteristic 0.41 nm lattice constant, corresponding to LEED patterns. Complimentary high-resolution synchrotron radiation photoemission spectroscopy measurements reveal the electronic structure of the honeycomb-like germanene sheet and further demonstrate that it overlays the surface of the Ag0.9Al0.1(111) alloy.

An approach for considering the object surface properties in a TLS stochastic model.

The interaction between laser beams and backscattering object surfaces lies at the fundamental working principle of any Terrestrial Laser Scanning (TLS) system. Optical properties of surfaces such as concrete, metals, wood, etc., which are commonly encountered in structural health monitoring of buildings and structures, constitute an important category of systematic and random TLS errors. This paper presents an approach for considering the random errors caused by object surfaces. Two surface properties are considered: roughness and reflectance. The effects on TLS measurements are modeled stepwise in form of a so-called synthetic variance-covariance matrix (SVCM) based on the elementary error theory. A line of work is continued for the TLS stochastic model by introducing a new approach for determining variances and covariances in the SVCM. Real measurements of cast stone façade elements of a tall building are used to validate this approach and show that the quality of the estimation can be improved with the appropriate SVCM.

Assignment games with population monotonic allocation schemes

AbstractWe characterize the assignment games which admit a population monotonic allocation scheme (PMAS) in terms of efficiently verifiable structural properties of the nonnegative matrix that induces the game. We prove that an assignment game is PMAS-admissible if and only if the positive elements of the underlying nonnegative matrix form orthogonal submatrices of three special types. In game theoretic terms it means that an assignment game is PMAS-admissible if and only if it contains either a veto player or a dominant veto mixed pair, or the game is a composition of these two types of special assignment games. We also show that in PMAS-admissible assignment games all core allocations can be extended to a PMAS, and the nucleolus coincides with the tau-value.

Revised Analytical Criteria for Aircraft Lateral - Directional Instabilities

Following the recently revised presentation of the theory of small-perturbation flight dynamics by the authors, the present paper considers the newly corrected aircraft lateral-directional equations with trim angle of attack as a parameter. The dynamics is modeled by a set of second-order equations written in matrix form which makes it convenient to derive conditions for onset of instability. The static instability criterion is directly obtained and is seen to match entirely the condition for spiral mode instability. An approximate criterion for dynamic instability is obtained by considering the undamped lateral-directional dynamics, which happens to be a Hamiltonian system. The onset of dynamic instability corresponds to the occurrence of a Hamiltonian Hopf bifurcation which can be explicitly evaluated. This yields a new criterion for dynamic instability which is simpler and easier to use than the Routh discriminant. Also, the new criterion has the Cnβ, dyn parameter as a leading term which explains why this parameter may be used as an indicator for wing rock onset.

A two‐stage parallel method on GPU based on hybrid‐compression‐format for diagonal matrix

AbstractSpMV (Sparse matrix‐vector multiplication) is an important computing core in traditional high‐performance computing and also one of the emerging data‐intensive applications. For diagonal sparse matrices, it is frequently necessary to fill in a large number of zeros to maintain the diagonal structure as for using DIA (Diagonal) storage format. The fact that filling with zeros may consume additional computing and memory resources, will certainly lead to degradation of the parallel computing performance of SpMV, further causing computing and storage redundancy. To solve the deficiencies of the DIA format, a Two‐stage parallel SpMV method is presented in this paper, which can reasonably distribute the data of diagonal matrix and irregular matrix to different CUDA kernels. As different corresponding compression methods are particularly designed for different matrix forms, a partition‐based hybrid format of DIA and CSR (HPDC) is therefore adopted in the two‐stage method to ensure load balancing among computing resources and continuity of data access on the diagonal. Simultaneously, a standard deviation among blocks is used as a criterion to obtain the optimal number of blocks and distribution of data. The experimental data were implemented in the Florida data set. Compared to DIA, cuSPARSE‐CSR, HDC, and BRCSD, the execution time of the Two‐stage method is shortened by 4, 3.4, 1.9, and 1.15, respectively.