Echelon Form Research Articles

Identification of Mathematics Prospective Teachers’ Conceptual Understanding in Determining Solutions of Linear Equation Systems

<p style="text-align:justify">This research is motivated by a linear equations system, which is the basis for studying necessary linear algebra materials, such as rank, range, linear independent/dependent, linear transformations, characteristic values and vectors. There are still prospective mathematics teachers who have difficulty solving linear equations system and understanding the form of row echelon and reduced row echelon forms. In this study, subjects were three prospective mathematics teachers from Swadaya Gunung Jati University Cirebon who were taking matrix algebra courses. This study aims to reveal the conceptual understanding of prospective mathematics teachers in determining the solution to systems of linear equations. The results show that there are still prospective mathematics teachers who only use memory about the properties and procedures in determining whether a matrix is said to be a row echelon form or a reduced row echelon form. Then, there is still weakness in building the algorithms' relationship due to the immature knowledge of the concepts. Researchers found that many prospective mathematics teachers were more comfortable solving problems that were performed procedurally. Further research is needed to determine how the mental construction process and mathematical conceptual knowledge of prospective mathematics teachers are through meaningful learning so that conceptual understanding is maximized.</p>

Gauss with Elementary Matrices in the SoftAge

Software can be employed to solve math problems and we have to cooperate with them not passively but in an active way. In this article, we present situations in which the software user should be careful in order to avoid the use of the straight commands of the software that may put a matrix in its echelon form or give straight its rank. Wrong results appear in Wolfram’s Mathematica software if the matrix has algebraic entries. Using the Gaussian elimination step by step method with the help of elementary matrices is a good strategy for the software user in such cases.

Algebraic Decoding of Quasi-Reversible BCH Codes Using Band Matrices

A Bose-Chaudhuri-Hocquenghem (BCH) is called quasi-reversible if there are consecutive elements a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,⋯,a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> in the defining set, where a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> is a negative integer and a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> is a positive integer. A matrix in band form presented in this article involves the use of the Newton identities and symmetric polynomial identities. This matrix is the coefficient matrix of a linear system. The solution to such a linear system is the coefficients of an error-locator polynomial for quasi-reversible BCH codes. Its computational complexity is significantly lower than BCH decoding using the linear system with coefficient matrix in column echelon form. This article also proposes a new partial syndrome matrix with radical locators and a lot of zeros. Such a matrix is still in band form. The expansion of its matrix determinant is exactly a radical-locator polynomial. Compared to the former radical-locator polynomials in the literature, the newly discovered radical-locator polynomial not only requires less storage memory but also results in computational benefits. Surprisingly, the resulting polynomials are much sparser than those for the narrow-sense BCH codes. Finally, a complete algebraic decoding algorithm for quasi-reversible BCH codes is provided. The error-locator and radical-locator polynomials of degree less than or equal to 5 are listed.

Performance assessment of multivariate process using time delay matrix

Researchers keep trying to find a way to reduce the requirement knowledge of Multivariate Process for performance assessments. Till now, the knowledge of interactor matrix or first several Markov parameter matrices are at least required to obtain the minimum variance benchmark in the performance assessment of Multivariate Process. In this paper, a novel minimum variance performance assessment technique is proposed for multivariate processes. Under the condition that the time delay indicator matrix can be written as row echelon form by row or/and column shift operations, the delay order is determined and the minimum variance (MV) benchmark can be straightforward obtained when the knowledge of time delay matrix is available. Comparing with the traditional approaches, the first several Markov parameter matrices and the knowledge of the plant are both not necessary required based on the proposed technique. It has been proved that the proposed technique can directly solve the problem of the performance assessment of Multivariate Process. The validity of the proposed algorithm will be verified through numerical examples, which include the practical industrial model - ‘Shell’ oil fractionator process.

Genetic algorithms with permutation-based representation for computing the distance of linear codes

Abstract Finding the minimum distance of linear codes is an NP-hard problem. Traditionally, this computation has been addressed by means of the design of algorithms that find, by a clever exhaustive search, a linear combination of some generating matrix rows that provides a codeword with minimum weight. Therefore, as the dimension of the code or the size of the underlying finite field increase, so it does exponentially the run time. In this work, we prove that, given a generating matrix, there exists a column permutation which leads to a reduced row echelon form containing a row whose weight is the code distance. This result enables the use of permutations as representation scheme, in contrast to the usual discrete representation, which makes the search of the optimum polynomial time dependent from the base field. In particular, we have implemented genetic and CHC algorithms using this representation as a proof of concept. Experimental results have been carried out employing codes over fields with two and eight elements, which suggests that evolutionary algorithms with our proposed permutation encoding are competitive with regard to existing methods in the literature. As a by-product, we have found and amended some inaccuracies in the Magma Computational Algebra System concerning the stored distances of some linear codes.

On Decoding Algebraic Codes Using Radical Locators

It is well-known that the decoding of algebraic codes with error locators have been extensively conceived in the literature over a half century. The radical locator, which is an $\omega $ th root of error locator, has recently been discovered. This paper focuses on the two classes of radical locators. The first is complete radical locators , where all the locators are assigned to radical locators. The second is partial radical locators in that there are only a few locators being radical locators. In the former case based on complete radical locators, a new square matrix whose determinant is a univariate radical-locator polynomial is proposed. In particular, this matrix is modified to allow a large square matrix. It can be transformed by Gaussian elimination to the matrix in a row-echelon form in which zeros appearing in the diagonal entries of the current matrix are able to determine errors implicitly. Furthermore, this work further extends the previous results from the univariate case to multivariate cases. The matrix methods found herein enable one to decode a large class of cyclic codes efficiently. In the latter case, the cyclotomic cosets are a practical approach to select a small subset of positive integers. They are employed to develop partial radical locators. Finally, in contrast with the complete radical locators, the algebraic decoding methods make a natural use of partial radical locators more widely and flexibly to arbitrary cyclic codes.

The C-matrix augmentation in a multi-route reaction mechanism

When the dimensional complexity of the chemical reaction mechanism arises, it needs special arrangements to deal with such issues. We propose an effective linear algebraic technique to split a complex reaction mechanism into different reaction completion routes to overcome this complexity. Here, three key matrices, the molecular, stoichiometric, and Horiuti matrices, are discussed in detail to deal with the whole mechanism as well as the different reaction routes. The C-matrix procedure is based on calculating the Reduced Row Echelon Form (RREF) for reducing each augmented matrix. This approach is efficient to find the typical results in chemical kinetics and engineering of chemical reactions, i.e., reaction routes, key, and non-key components, key and non-key elements, dependent and independent reaction steps, and overall reaction free from the catalyst. This procedure is very useful to develop new software tools for analyzing complex chemical networks.

Two algorithms for the exchange lemma

In this note, we present two new algorithms for the Steinitz Exchange Lemma. They are grounded on a single application of a procedure (finding either a row echelon form or a basic minor) that has to be applied repeatedly in previously known algorithms. Proving correctness of both the algorithms, we obtain two new, direct proofs of the Exchange Lemma.

Two-sided linear matrix inequality solution of affine input matrix for feasible discrete finite-time sliding mode control of uncertain nonlinear mechanical machines

In this article, a new control method is proposed based on finite-time discrete sliding mode control for uncertain multi-input multi-output systems which are affine to their inputs considering uncertain input multipliers in the case where signs of input gains remain constant over uncertainty spaces. In addition, a method for solving a set of convex control inequalities is introduced. The proposed control strategy is based on merging data obtained from investigation of common candidate Lyapunov functions assigned to various subsystems and their subsequent decoupling based on matrix elementary row operations. Initially, separate sliding functions corresponding to a single degree of freedom are assigned to each subsystem in the overall multi-input multi-output system, which results in obtaining a convex inequality corresponding to input bounds. Stacking the data obtained from various subsystems, the product of the uncertain input gain matrix in input vector is obtained as the middle term in a set of convex inequalities. Subsequently, the convex inequality is solved according to a set of matrix elementary row operations transforming the corresponding input matrix to row echelon form such that the bounds of each input are clearly expressed. Then, based on assigning input bounds proximity factors to each lower bound–upper bound duo, appropriate control inputs are generated. Chattering effects are eliminated as no switching term is included in construction of the control model. Effectiveness of the proposed method is demonstrated using numerical simulations. The implementation of control algorithm using microprocessors is also illustrated, indicating the feasibility of digital application.

A constrained force density method for the funicular analysis and design of arches, domes and vaults

A computational approach is proposed to cope with the assessment/design of arcuated structures through funicular analysis. As investigated in the literature, the equilibrium of funicular networks can be conveniently handled through the force density method, using independent sets of branches in the case of networks with fixed plan geometry. In this contribution, the minimization/maximization of the horizontal thrusts of funicular networks is stated both in terms of any set of independent force densities and of the height of the restrained nodes. The dependent force densities are written as a function of the independent ones by handling the horizontal equilibrium through the reduced row echelon form of its augmented matrix. A suitable set of local constraints is formulated to prescribe lower and upper bounds for the vertical coordinates of both restrained and unrestrained vertices of the network, and to enforce no-tension members. Due to its peculiar form, this problem can be efficiently solved through techniques of sequential convex programming that were originally conceived to handle multi–constrained formulations of size optimization for elastic structures. Both vertical and horizontal loads are considered to assess the predictions of the proposed algorithm that can handle networks with general topology and boundary conditions.

Systematic encoders for generalized Gabidulin codes and the q-analogue of Cauchy matrices

We characterize the generator matrix in standard form of generalized Gabidulin codes. The parametrization we get for the non-systematic part of this matrix coincides with the q-analogue of generalized Cauchy matrices, leading to the definition of q-Cauchy matrices. These matrices can be represented very conveniently and their representation allows to define new interesting subfamilies of generalized Gabidulin codes whose generator matrix is a structured matrix. In particular, as an application, we construct Gabidulin codes whose generator matrix is the concatenation of an identity block and a Toeplitz/Hankel matrix. In addition, our results allow to give a new efficient criterion to verify whether a rank metric code of dimension k and length n is a generalized Gabidulin code. This criterion is only based on the computation of the rank of one matrix and on the verification of the linear independence of two sets of elements and it requires O(m⋅F(k,n)) field operations, where F(k,n) is the cost of computing the reduced row echelon form of a k×n matrix. Moreover, we also provide a characterization of the generator matrix in standard form of general MRD codes.

The echelon Markov and Chebyshev inequalities

A multivariate version of the sharp Markov inequality is derived, when associated probabilities are extended to segments of the supports of non-negative random variables, where the probabilities take echelon forms. It is shown that when some positive lower bounds of these probabilities are available, the multivariate Markov inequality without the echelon forms is improved. The corresponding results for Chebyshev’s inequality are also obtained.

Analysis of Line Outage Detection in Nigeria 330kV Transmission Lines using Phasor Measurement Units

In this work, an analysis of line outage detection in Nigeria 330kV transmission lines using Phasor Measurement Units was presented. This requires collection and analysis of the data obtained from Transmission Company of Nigeria with the aid of PSAT 2.10.1 / MATLAB SIMULINK using Newton-Raphson power flow algorithm and also to determine the effectiveness of PMU when introduced in our power system network. 12 buses and 3 Generators system were considered for the studied. This was achieved by collecting relevant transmission parameters for 330kV line and was simulated on PSAT 2.10.1 and MATLAB 2015a using Newton-Raphson power flow algorithm. The work involved an offline and online analysis. For the offline analysis the admittance / impedance matrix for Y-bus and bus voltage for pre-outage was obtained via the power flow analysis and change in impedance for the lines were calculated. These values were further normalised in order to reduce the value to a row echelon form. Then for the online analysis; the change in phase angle from the Phasor Measurement Unit (PMU) online simulation for pre-outage and also post-outage was calculated and a normalised column matrix was gotten. Finally, the effectiveness of the line outage detection was graphically represented using MATLAB software to plot the values of the normalised values of the offline and online analysis; i.e., by comparing the normalised form of the offline and online values. These results clearly show that PMUs gives an accurate monitoring and total observability when introduced in Nigeria power system.

Model specification and selection for multivariate time series

Three major difficulties are identified with an established echelon form approach (see Hannan (1987)) to specifying a Vector Autoregressive Moving Average, VARMA, model for an observed time series. A family of state space representations, valid for each integer, h, is introduced, and collectively referred to as multistep state space representations. This family includes as its special case, with h=0, a state space representation introduced earlier by Akaike (1974), and, with h=1, that introduced by Cooper and Wood (1982). Appropriate generalizations of the notions of minimality, McMillan degree, left matrix fraction description and Kronecker indices, as applicable individually to each member of this family, are presented. The reverse echelon form and state space representation corresponding to the Kronecker indices for each h are derived, and the former illustrated with three examples of standard VARMA processes. The question of how the presence of zero constraints on the coefficients of a reverse echelon form may be detected solely from an inspection of the Kronecker indices is examined. A canonical correlation procedure proposed originally by Akaike (1976) for h=0 is considered for estimating the Kronecker indices with each h. The efficacy of the estimation procedure is investigated by a simulation study. A procedure is suggested for implementing the new approach introduced in this paper with an observed time series, and three different applications of this approach are outlined. This approach is also related to some of its alternatives, including the Kronecker invariants of Poskitt (1992) and the scalar component approach of Tiao and Tsay (1989).

Groebner bases algorithm for optimal PMU placement

In this paper a new method for solving optimal phasor measurement units (PMU) placement problem for full observability in power systems is proposed. A new algorithm is based on forming and solving Groebner bases set of equations for a defined optimization problem. The technique of Groebner bases is applicable wherever the problem may be presented by the set of polynomials and as such, it can be found in a variety of applied mathematics problems. Groebner bases of the set of polynomials represents the canonical form equivalent to the original set, similar to the way the row echelon form for linear systems represents the triangular form of the linear equations set. The canonical form of the Groebner bases polynomials, along with the utilization of recursive algorithms, allows the solution to be effortlessly obtained. Optimal phasor measurement units placement (OPP) problem can be presented in a linear or nonlinear polynomial form, both of which are used in this paper. The settings of OPP problem have been proposed in this paper with the inclusion of zero injection buses (ZIBs), in normal system states and considering contingency related to PMU outage. The problem formulation is defined according to the application of Groebner bases technique. The proposed model is capable of obtaining the full set of optimal solutions instead of only one or partial solutions. The efficiency of the proposed method is tested on IEEE 14-bus, IEEE 24-bus, IEEE 30-bus, IEEE 118-bus and Polish 2383-bus test system and the results are compared with those of some recently reported methods.

A module-theoretic approach to matroids

Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces as modules over an idempotent semifield. All together, this provides bridges between the combinatorics of matroids, the algebra of idempotent modules, and the geometry of tropical linear spaces. The goal of this paper is to strengthen and expand these bridges by systematically developing the idempotent module theory of matroids. Applications include a geometric interpretation of strong matroid maps and the factorization theorem; a generalized notion of strong matroid maps, via an embedding of the category of matroids into a category of module homomorphisms; a monotonicity property for the stable sum and stable intersection of tropical linear spaces; a novel perspective of fundamental transversal matroids; and a tropical analogue of reduced row echelon form.

Minimum distance computation of linear codes via genetic algorithms with permutation encoding

We design a heuristic method, a genetic algorithm, for the computation of an upper bound of the minimum distance of a linear code over a finite field. By the use of the row reduced echelon form, we obtain a permutation encoding of the problem, so that its space of solutions does not depend on the size of the base field or the dimension of the code. Actually, the efficiency of our method only grows non-polynomially with respect to the length of the code.

Optimising the load path of compression-only thrust networks through independent sets

This paper presents network load path optimisation for the weight minimisation of compression-only thrust networks, allowing for the design of material efficient surface structures. A hybrid evolutionary and function-gradient optimisation process finds the optimal internal force state of the network, by manipulating the force densities of a selected number of edges based on the network indeterminacy. These selected edges are the independent sets, and are found through the Reduced Row Echelon form of the network’s equilibrium matrix. It was found that networks can have certain independent sets that have a significant influence on both the stability of the optimisation algorithm, and in the final load path/volume of the structure. Finding the most effective independent sets was handled by data-driven methods, applied to many thousands of independent set trials. This provided insight into the behaviour of the underlying network and dramatically increased the rate of finding successful independent sets. The importance and weights of the network edges highlighted key areas of the network that allowed structural judgement and improvements to be made.

Анализ сингулярных конфигураций мобильного манипулятора KUKA youBot

The article considers the KUKA youBot mobile manipulator with open software, which is intended for research and education purposes. Singular configurations of this apparatus have not been described in the literature as yet. The KUKA youBot system is a controlled platform, on which a five-axis manipulator with a grip is mounted. The platform can be moved in the horizontal plane by means of four mecanum wheels. The robot position in the absolute space is determined by eight generalized coordinates. As is known, the KUKA youBot inverse kinematics problem allows its exact analytical solution to be obtained. A rectangular Jacobi matrix for mapping the space of the robot generalized coordinates into the space of its end effector location parameters is constructed. The obtained Jacobi matrix is transformed to an equivalent echelon form by means of elementary operations. An analysis of the echelon matrix entries carried out using analytical tools has shown that, whatever the locations of the KUKA youBot system’s platform and the system arm first, fourth and fifth links, there is a singular configuration in which the robot's arm second and third links are stretched along the vertical line. In this case, the Jacobi matrix rank becomes equal to five (less than its full rank equal to six). When being in the above-mentioned singular configuration, the KUKA youBot system end effector cannot perform some of the movements the pattern of which is described in the presented paper.

Simplified Attitude Determination Algorithm Using Accelerometer and Magnetometer with Extremely Low Execution Time

Accelerometer-magnetometer attitude determination is a common and vital medium processing technique in industrial robotics and consumer electronics. In this paper, we report a novel analytic attitude solution to the accelerometer-magnetometer combination in the sense of Wahba’s problem. The Davenport matrix is analytically given and its eigenvalues are computed. Through derivations, the eigenvalues are simplified to very short expressions. Then, the corresponding eigenvectors are given accordingly via matrix row operations. The system is highly optimized based on the factorization and simplification of the obtained row-echelon form, which makes it computationally fast in practice. In this way, it is named as the fast accelerometer-magnetometer combination (FAMC). Experiments on the correctness and advantages of the proposed solution are conducted. The results show that compared with conventional solutions, the proposed analytic solution is not only correct and accurate, but to our knowledge, the most time efficient as well.