Two-sided linear split quaternionic equations with n unknowns

In this paper, we investigate linear split quaternionic equations with the terms of the form axb. We give a new method of solving general linear split quaternionic equations with one, two and n unknowns. Moreover, we present some examples to show how this procedure works.     

Cramer's rule for quaternionic systems of linear equations

New definitions of determinant functionals over the quaternion skew field are given in this paper. The inverse matrix over the quaternion skew field is represented by analogues of the classical adjoint matrix. Cramer's rules for right and left quaternionic systems of linear equations have been obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 67–94, 2007.     

On the determinant of quaternionic polynomial matrices and its application to system stability

In this paper, we propose a definition of determinant for quaternionic polynomial matrices inspired by the well‐known Dieudonné determinant for the constant case. This notion allows to characterize the stability of linear dynamical systems with quaternionic coefficients, yielding results which generalize the ones obtained for the real and complex cases. Copyright © 2007 John Wiley & Sons, Ltd.     

关于齐次四元Monge-Ampère方程的一些注记北大核心CSCD

本文考虑了四元数空间Hn中齐次四元Monge-Ampère方程的狄利克雷问题解的正则性.首先,当区域是边界为C1,1的强拟凸域时,作者给出了解的Lipschitz估计.其次,考虑了四元MongeAmpère算子的收敛性.最后,讨论了齐次四元Monge-Ampère方程的粘性次解与F-次调和函数之间的关系.     

Quaternionic spherical wave functions

The scalar spherical wave functions (SWFs) are solutions to the scalar Helmholtz equation obtained by the method of separation of variables in spherical polar coordinates. These functions are complete and orthogonal over a sphere, and they can, therefore, be used as a set of basis functions in solving boundary value problems by spherical wave expansions. In this work, we show that there exists a theory of functions with quaternionic values and of three real variables, which is determined by the Moisil–Theodorescu‐type operator with quaternionic variable coefficients, and which is intimately related to the radial, angular and azimuthal wave equations. As a result, we explain the connections between the null solutions of these equations, on one hand, and the quaternionic hyperholomorphic and anti‐hyperholomorphic functions, on the other. We further introduce the quaternionic spherical wave functions (QSWFs), which refine and extend the SWFs. Each function is a linear combination of SWFs and products of ‐hyperholomorphic functions by regular spherical Bessel functions. We prove that the QSWFs are orthogonal in the unit ball with respect to a particular bilinear form. Also, we perform a detailed analysis of the related properties of QSWFs. We conclude the paper establishing analogues of the basic integral formulae of complex analysis such as Borel–Pompeiu's and Cauchy's, for this version of quaternionic function theory. As an application, we present some plot simulations that illustrate the results of this work. Copyright © 2016 John Wiley & Sons, Ltd.     

Real matrix representations of complex split quaternions with applications

In this paper, we introduce 8×8 real matrix representations of complex split quaternions. Then, the relations between real matrix representations of split and complex split quaternions are stated. Moreover, we investigate some linear split and complex split quaternionic equations with split Fibonacci and complex split Fibonacci quaternion coefficients. Finally, we also give some numerical examples as applications of real matrix representation of complex split quaternions.     

Normal extensions of subnormal linear relations via quaternionic Cayley transforms

A quaternionic Cayley transform for linear relations is introduced and some of its properties are exhibited. We emphasize the role played by the linear relations whose quaternionic Cayley transforms are unitary operators, which happen to be normal relations, and investigate the class of those linear relations which extend to such normal relations.     

Sharper uncertainty principles in quaternionic Hilbert spaces

The uncertainty principle for quaternionic linear operators in quaternionic Hilbert spaces is established, which generalizes the result of Goh-Micchelli. It turns out that there appears an additional term given by a commutator that reflects the feature of quaternions. The result is further strengthened when one operator is self-adjoint, which extends under weaker conditions the uncertainty principle of Dang-Deng-Qian from complex numbers to quaternions. In particular, our results are applied to concrete settings related to quaternionic Fock spaces, quaternionic periodic functions, quaternion Fourier transforms, quaternion linear canonical transforms, and nonharmonic quaternion Fourier transforms.     

Complements to full order observer design for linear systems with unknown inputs

This note completes the results obtained in [M. Darouach, M. Zasadzinski, S.J. Xu, Full-order observers for linear systems with unknown inputs, IEEE Trans. Automat. Control., AC-39 (3) (1994) 606–609]. It presents a full order observer for linear systems with unknown inputs in the state and in the measurement equations. It gives a more general approach for the observer’s design than that presented in the above mentioned reference and it shows that all the obtained results are independent of the choice of the generalized inverses considered in the observer’s design. Continuous- and discrete-time systems are considered.     

Lyapunov function,parameter estimation,synchronization and chaotic cryptography

In this paper we have investigated a new method of synchronization between two non linear systems based upon lyapunov function and parameter estimation through modulational equations. The driving and response systems, both are different and their parameters are unknown.We have constructed the parameter modulation equations and control laws to achieve the synchronization. This method is well applied to two different three dimensional systems and the transverse lyapunov exponents show the effectiveness of the method. Further more we have investigated the cryptographical applications with the help of the above two systems.     

Heights and Degrees: A Note on a Paper of C. Liebendörfer and G. Rémond

We verify in the rational quaternionic case a conjecture of C. Liebend?rfer relating the degrees of abelian subvarieties of certain polarized abelian varieties to the (quaternionic) heights of their defining equations.     

Dirichlet problem of quaternionic Monge–Ampère equations

In this paper, the author studies quaternionic Monge–Ampère equations and obtains the existence and uniqueness of the solutions to the Dirichlet problem for such equations without any restriction on domains. Our paper aims to answer the question proposed by Semyon Alesker in [3]. It also extends relevant results in [8] to the quaternionic vector space.     

Numerical Solution of Inverse Problems for a Hyperbolic Equation with a Small Parameter Multiplying the Highest Derivative

We consider two inverse problems for a hyperbolic equation with a small parameter multiplying the highest derivative. The inverse problems are reduced to systems of linear Volterra integral equations of the second kind for the unknown functions. These systems are used to prove the existence and uniqueness of the solution of the inverse problems and numerically solve them. The applicability of the methods developed here to the approximate solution of the problem on an unknown source in the heat equation is studied numerically.     

On the continuous dependence of solutions of linear systems of differential-algebraic equations on a parameter

We consider linear systems of ordinary differential equations with identically degenerate matrix multiplying the derivative of the unknown vector function. The matrices specifying the system are assumed to depend on a parameter. We obtain criteria for the continuous dependence of the solutions of the system on the parameter and the asymptotic equivalence of solutions of the original and perturbed systems.     

On quaternionic analysis for the Schrödinger operator with a particular potential and its relation with the Mathieu functions

It has been found recently that there exists a theory of functions with quaternionic values and in two real variables, which is determined by a Cauchy–Riemann‐type operator with quaternionic variable coefficients, and that is intimately related to the so‐called Mathieu equations. In this work, it is all explained as well as some basic facts of the arising quaternionic function theory. We establish analogues of the basic integral formulas of complex analysis such as Borel–Pompeiu's, Cauchy's, and so on, for this version of quaternionic function theory. This theory turns out to be in the same relation with the Schrödinger operator with special potential as the usual holomorphic functions in one complex variable, or quaternionic hyperholomorphic functions, or functions of Clifford analysis, are with the corresponding Laplace operator. Moreover, it is similar to that of α‐hyperholomorphic functions and the Helmholtz operator. Copyright © 2012 John Wiley & Sons, Ltd.     

Geometric theory of differential systems: Linearization criterion for systems of second-order ordinary differential equations with a 4-dimensional solvable symmetry group of the Lie–Petrov type VI 1

In the framework of projective-geometric theory of systems of differential equations developed by the authors, this paper studies the group properties of systems of two (resolved with respect to the second derivatives) second-order ordinary differential equations whose right-hand sides are polynomials of the third degree with respect to the derivatives of the unknown functions. A classification of such systems admitting four-dimensional symmetry group of the Lie–Petrov type VI ₁ is given. For each of the systems, a necessary and sufficient linearization criterion is obtained, i.e., the authors find the necessary and sufficient conditions under which, by a change of variables, the system can be reduced to a differential system whose integral curves are straight lines and are expressed by three linear parametric equations or two linear equations with constant coefficients. For all linearizable systems, the linearizing changes of variables are indicated. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 54, Suzdal Conference–2006, Part 2, 2008.     

Optimal control of singular systems via piecewise linear polynomial functions

A method for finding the optimal control of linear singular systems with a quadratic cost functional using piecewise linear polynomial functions is discussed. The state variable, state rate, and the control vector are expanded in piecewise linear polynomial functions with unknown coefficients. The relation between the coefficients of the state rate with state variable is provided and the necessary condition of optimality is derived as a linear system of algebraic equations in terms of the unknown coefficients of the state and control vectors. A numerical example is included to demonstrate the validity and the applicability of the technique. Copyright © 2002 John Wiley & Sons, Ltd.     

Quaternionic Monge-Ampère equations

The main result of this article is the existence and uniqueness of the solution of the Dirichlet problem for quaternionic Monge-Ampère equations in quaternionic strictly pseudoconvex bounded domains in ℍ ⁿ . We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2].     

Quaternionic triangular linear operators

Triangular operators are an essential tool in the study of nonselfadjoint operators that appear in different fields with a wide range of applications. Although the development of a quaternionic counterpart for this theory started at the beginning of this century, the lack of a proper spectral theory combined with problems caused by the underlying noncommutative structure prevented its real development for a long time. In this paper, we give criteria for a quaternionic linear operator to have a triangular representation, namely, under which conditions such operators can be represented as a sum of a diagonal operator with a Volterra operator. To this effect, we investigate quaternionic Volterra operators based on the quaternionic spectral theory arising from the S-spectrum. This allow us to obtain conditions when a non-selfadjoint operator admits a triangular representation.