Application of the theory of linear algebra on mathematical modeling practice

The application level of mathematics in each science discipline signs the level of development of this science. With the advancement of science and technology, especially the rapid development of computer technology, mathematics has permeated from natural scientific technology to agricultural construction, from economic activities to all areas of social life. Generally, when the actual problem requires us to provide quantitative results of analysis, forecasting, decision making, control and other aspects for real object under study, we are often inseparable from the application of mathematics. Mathematical modeling is the key to this process, whose purpose is to make mathematics applied to social and social services, and using mathematics to solve practical problems is through mathematical models. When using mathematical methods to solve some practical problems, we usually first transfer practical problems into mathematical language, and then abstract them into a mathematical model.

Research on Factors of Linear Algebra Learning Effect

Linear algebra has a very important application in physics and technical disciplines. This article conducted a questionnaire survey on the factors that affect the effect of linear algebra learning;the questionnaire contains several aspects of learning attitude, learning interest, learning methods, teaching methods, etc.;based on recycling data, cross chi-square test and multiple logistic regression analysis are used to obtain the factors that affect the effect of linear algebra learning. The research results show that: learning methods, learning attitudes, teaching methods and elementary algebra basics are the main factors that affect the learning effect of linear algebra;among them, there are positive correlations between teaching methods, learning methods, learning attitudes and learning effects;teaching methods, learning methods 3. The three principal components of learning attitude are positively correlated. Based on the research and analysis, the following conclusions are drawn: finding a suitable learning method for the college students and maintaining a positive learning attitude are effective means to improve the linear algebra learning effect of the college students;in teaching, it is recommended to advance with the times, the teaching content and teaching methods innovate to stimulate students’ interest in learning, thus improving the learning effect of college students’ linear algebra courses.

On Linear Algebra for Non Mathematics Majors

To find out what knowledge in linear algebra is essential to non-mathematics students, a reverse tracking method was used. Based on practical problems likely to encountered by students in subsequent engineering courses, the minimum contents required has been determined. Rules are proposed to meet the background of most freshman students. An application oriented, easy to understand, computer based text book “Applied Popular Linear Algebra with MATLAB” [1] was published.

Inverse problem in linear algebra

With the increasingly widespread application of linear algebra theory, and in its opposite direction is not enough emphasis, linear algebra, several important points： matrix, determinant, linear equations, linear transformations, matrix keratosis and other anti-deepening understanding of the basics and improve the comprehensive ability to solve problems.

The Continuous Analogy of Newton’s Method for Solving a System of Linear Algebraic Equations

We propose a continuous analogy of Newton’s method with inner iteration for solving a system of linear algebraic equations. Implementation of inner iterations is carried out in two ways. The former is to fix the number of inner iterations in advance. The latter is to use the inexact Newton method for solution of the linear system of equations that arises at each stage of outer iterations. We give some new choices of iteration parameter and of forcing term, that ensure the convergence of iterations. The performance and efficiency of the proposed iteration is illustrated by numerical examples that represent a wide range of typical systems.

Variational algorithms for linear algebra

Quantum algorithms have been developed for efficiently solving linear algebra tasks.However,they generally require deep circuits and hence universal fault-tolerant quantum computers.In this work,we propose variational algorithms for linear algebra tasks that are compatible with noisy intermediate-scale quantum devices.We show that the solutions of linear systems of equations and matrix–vector multiplications can be translated as the ground states of the constructed Hamiltonians.Based on the variational quantum algorithms,we introduce Hamiltonian morphing together with an adaptive ans?tz for efficiently finding the ground state,and show the solution verification.Our algorithms are especially suitable for linear algebra problems with sparse matrices,and have wide applications in machine learning and optimisation problems.The algorithm for matrix multiplications can be also used for Hamiltonian simulation and open system simulation.We evaluate the cost and effectiveness of our algorithm through numerical simulations for solving linear systems of equations.We implement the algorithm on the IBM quantum cloud device with a high solution fidelity of 99.95%.

On Implementing the Symbolic Preprocessing Function over Boolean Polynomial Rings in Gr?bner Basis Algorithms Using Linear Algebra

Some techniques using linear algebra was introduced by Faugore in F4 to speed up the reduction process during Grobner basis computations. These techniques can also be used in fast implementations of F5 and some other signature-based Grobner basis algorithms. When these techniques are applied, a very important step is constructing matrices from critical pairs and existing polynomials by the Symbolic Preprocessing function （given in F4）. Since multiplications of monomials and polynomials are involved in the Symbolic Preprocessing function, this step can be very costly when the number of involved polynomials/monomials is huge. In this paper, multiplications of monomials and polynomials for a Boolean polynomial ring are investigated and a specific method of implementing the Symbolic Preprocessing function over Boolean polynomial rings is reported. Many examples have been tested by using this method, and the experimental data shows that the new method is very efficient.

Parallel Implementation of Linear Algebra Problems on Dawning-1000

In this paper, some parallel algorithms are described for solving numerical linear algebra problems on Dawning-1000. They include matrix multiplication, LU factorization of a dense matrix, Cholesky factorization of a symmetric matrix, and eigendecomposition of symmetric matrix for real and complex data types. These programs are constructed based on fast BLAS library of Dawning-1000 under NX environment.Some comparison results under different parallel environments and implementing methods are also given for Cholesky factorization. The execution time, measured performance and speedup for each problem on Dawning-1000 are shown. For matrix multiplication and LU factorization, 1.86GFLOPS and 1.53GFLOPS are reached.

GENERALIZED DERIVATIONS ON PARABOLIC SUBALGEBRAS OF GENERAL LINEAR LIE ALGEBRAS

Let P be a parabolic subalgebra of a general linear Lie algebra gl（n,F） over a field F, where n ≥ 3, F contains at least n different elements, and char（F） ≠ 2. In this article, we prove that generalized derivations, quasiderivations, and product zero derivations of P coincide, and any generalized derivation of P is a sum of an inner derivation, a central quasiderivation, and a scalar multiplication map of P. We also show that any commuting automorphism of P is a central automorphism, and any commuting derivation of P is a central derivation.

Invertible Linear Maps on the General Linear Lie Algebras Preserving Solvability

Let Mn be the algebra of all n × n complex matrices and gl（n, C） be the general linear Lie algebra, where n ≥ 2. An invertible linear map φ ： gl（n, C） → gl（n, C） preserves solvability in both directions if both φ and φ-1 map every solvable Lie subalgebra of gl（n, C） to some solvable Lie subalgebra. In this paper we classify the invertible linear maps preserving solvability on gl（n, C） in both directions. As a sequence, such maps coincide with the invertible linear maps preserving commutativity on Mn in both directions.

3D Object Recognition Based on Linear Lie Algebra Model

A surface model called the fibre bundle model and a 3D object model based on linear Lie algebra model are proposed. Then an algorithm of 3D object recognition using the linear Lie algebra models is presented. It is a convenient recognition method for the objects which are symmetric about some axis. By using the presented algorithm, the representation matrices of the fibre or the base curve from only finite points of the linear Lie algebra model can be obtained. At last some recognition results of practicalities are given.

Non-linear Invertible Maps that Preserve Staircase Subalgebras

Let g be the general linear Lie algebra consisting of all n x n matrices over a field F and with the usual bracket operation {x, y} =xy - yx. An invertible map φ ： g →g is said to preserve staircase subalgebras if it maps every staircase subalgebra to some staircase subalgebra of the same dimension. In this paper, we devote to giving an explicit description on the invertible maps on g that preserve staircase subalgebras.

LINEAR *-DERIVATIONS ON JB*-ALGEBRAS

It is shown that for a derivation on a JB* -algebra B, there exists a unique C-linear *-derivation D:B→B near the derivation.

THE LARGEST SOLUTION OF LINEAR EQUATION OVER THE COMPLETE HEYTING ALGEBRA

Let （L, 〈, V, A） be a complete Heyting algebra. In this article, the linear system Ax = b over a complete Heyting algebra, where classical addition and multiplication operations are replaced by V and A respectively, is studied. We obtain： （i） the necessary and sufficient conditions for S（A,b）≠Ф; （ii） the necessary conditions for IS（A,b）｜ = 1. We also obtain the vector x ∈ Ln and prove that it is the largest element of S（A, b） if S（A, b）≠Ф.

Non-Linear Semi-Quantum Hamiltonians and Its Associated Lie Algebras

We show that the non-linear semi-quantum Hamiltonians which may be expressed as(whereis the set of generators of some Lie algebra and are the classical conjugated canonical variables) always close a partial semi Lie algebra under commutation and, because of this, it is always possible to integrate the mean values of the quantum degrees of freedom of the semi-quantum non-linear system in the fashion:(whereis the Maximum Entropy Principle density operator) and, so, these kind of Hamiltonians always have associated dynamic invariants which are expressed in terms of the quantum degrees of freedom’s mean values. Those invariants are useful to characterize the kind of dynamics (regular or irregular) the system displays given that they can be fixed by means of the initial conditions imposed on the semi-quantum non-linear system.

Non-linear Maps on Borel Subalgebras of Simple Lie Algebras Preserving Abelin Ideals

Let g be a complex simple Lie algebra of rank ι, b the standard Borel subalgebra. An invertible map on Ь is said to preserve abelian ideals if it maps each abelian ideal to some such ideal of the same dimension. In this article, by using some results of Chevalley groups, the theory of root systems and root space decomposition, the author gives an explicit description on such maps of Ь.

THE SOLUTION FOR THE GENERALIZED RICCATIALGEBRAIC EQUATIONS OF LINEAR EQUALITY CONSTRAINT SYSTEM

Based on the dynamic equation, the performance functional and the system constraint equation of time-invariant discrete LQ control problem, the generalized Riccati equations of linear equality constraint system are obtained according to the minimum principle, then a deep discussion about the above equations is given, and finally numerical example is shown in this paper.

Robust entry guidance using multi-segment linear pseudospectral model predictive control

This paper presents a robust entry guidance algorithm for the high lift-to-drag ratio entry vehicle that employs the recently developed pseudospectral model predict control in a segmented manner. Here, the guidance commands are the longitudinal lift-to-drag (L/D) and bank reversal commands, which are calculated by successively solving multiple segment linear algebraic equations. These equations are derived using the linear pseudospectral method, control parametrization and calculus of variations. The method uses orthogonal polynomials and computes the updates through a series of analytical formulae, which makes it accurate and computationally effective. Moreover, it is able to adjust the number of bank reversals by providing the precise bank reversal point so as to fully exploit the potential of lateral maneuver. The method also employs proportional navigation and polynomial guidance after the last bank reversal to meet multiple terminal constraints. High-fidelity numerical simulations with various destinations are carried out to demonstrate its applicability. Furthermore, Monte Carlo simulations are also conducted to show that the proposed algorithm consistently offers very stable and robust performances and has superior performances in computational efficiency, guidance accuracy and lateral trajectory shaping capability in comparison with other typical methods. © 1990-2011 Beijing Institute of Aerospace Information.

Robust H_∞ Control for Uncertain Markovian Jump Linear Time-Delay Systems

This paper studies the robust stochastic stabilization and robust H∞ control for linear time-delay systems with both Markovian jump parameters and unknown norm-bounded parameter uncertainties. This problem can be solved on the basis of stochastic Lyapunov approach and linear matrix inequality (LMI) technique. Sufficient conditions for the existence of stochastic stabilization and robust H∞ state feedback controller are presented in terms of a set of solutions of coupled LMIs. Finally, a numerical example is included to demonstrate the practicability of the proposed methods.

The Application of Linear Ordinary Differential Equations

In this article, we will explore the applications of linear ordinary differential equations (linear ODEs) in Physics and other branches of mathematics, and dig into the matrix method for solving linear ODEs. Although linear ODEs have a comparatively easy form, they are effective in solving certain physical and geometrical problems. We will begin by introducing fundamental knowledge in Linear Algebra and proving the existence and uniqueness of solution for ODEs. Then, we will concentrate on finding the solutions for ODEs and introducing the matrix method for solving linear ODEs. Eventually, we will apply the conclusions we’ve gathered from the previous parts into solving problems concerning Physics and differential curves. The matrix method is of great importance in doing higher dimensional computations, as it allows multiple variables to be calculated at the same time, thus reducing the complexity.