1,130 research outputs found
Roth’s solvability criteria for the matrix equations AX - XB^ = C and X - AXB^ = C over the skew field of quaternions with aninvolutive automorphism q ¿ qˆ
The matrix equation AX-XB = C has a solution if and only if the matrices A C 0 B and A 0
0 B are similar. This criterion was proved over a field by W.E. Roth (1952) and over the skew field of quaternions by Huang Liping (1996). H.K. Wimmer (1988) proved that the matrix equation X - AXB = C over a field has a solution if and only if the matrices A C 0 I and I 0 0 B are simultaneously equivalent to A 0 0 I and
I 0 0 B . We extend these criteria to the matrix equations AX- ^ XB = C and X - A ^ XB = C over the skew field of quaternions with a fixed involutive automorphism q ¿ ˆq.Postprint (author's final draft
Tele-Autonomous control involving contact
Object localization and its application in tele-autonomous systems are studied. Two object localization algorithms are presented together with the methods of extracting several important types of object features. The first algorithm is based on line-segment to line-segment matching. Line range sensors are used to extract line-segment features from an object. The extracted features are matched to corresponding model features to compute the location of the object. The inputs of the second algorithm are not limited only to the line features. Featured points (point to point matching) and featured unit direction vectors (vector to vector matching) can also be used as the inputs of the algorithm, and there is no upper limit on the number of the features inputed. The algorithm will allow the use of redundant features to find a better solution. The algorithm uses dual number quaternions to represent the position and orientation of an object and uses the least squares optimization method to find an optimal solution for the object's location. The advantage of using this representation is that the method solves for the location estimation by minimizing a single cost function associated with the sum of the orientation and position errors and thus has a better performance on the estimation, both in accuracy and speed, than that of other similar algorithms. The difficulties when the operator is controlling a remote robot to perform manipulation tasks are also discussed. The main problems facing the operator are time delays on the signal transmission and the uncertainties of the remote environment. How object localization techniques can be used together with other techniques such as predictor display and time desynchronization to help to overcome these difficulties are then discussed
Algebraic technique for mixed least squares and total least squares problem in the reduced biquaternion algebra
This paper presents the reduced biquaternion mixed least squares and total
least squares (RBMTLS) method for solving an overdetermined system in the reduced biquaternion algebra. The RBMTLS method is suitable when
matrix and a few columns of matrix contain errors. By examining real
representations of reduced biquaternion matrices, we investigate the conditions
for the existence and uniqueness of the real RBMTLS solution and derive an
explicit expression for the real RBMTLS solution. The proposed technique covers
two special cases: the reduced biquaternion total least squares (RBTLS) method
and the reduced biquaternion least squares (RBLS) method. Furthermore, the
developed method is also used to find the best approximate solution to over a complex field. Lastly, a numerical example is presented to
support our findings.Comment: 19 pages, 3 figure
L-structure least squares solutions of reduced biquaternion matrix equations with applications
This paper presents a framework for computing the structure-constrained least
squares solutions to the generalized reduced biquaternion matrix equations
(RBMEs). The investigation focuses on three different matrix equations: a
linear matrix equation with multiple unknown L-structures, a linear matrix
equation with one unknown L-structure, and the general coupled linear matrix
equations with one unknown L-structure. Our approach leverages the complex
representation of reduced biquaternion matrices. To showcase the versatility of
the developed framework, we utilize it to find structure-constrained solutions
for complex and real matrix equations, broadening its applicability to various
inverse problems. Specifically, we explore its utility in addressing partially
described inverse eigenvalue problems (PDIEPs) and generalized PDIEPs. Our
study concludes with numerical examples.Comment: 30 page
Heights and quadratic forms: on Cassels' theorem and its generalizations
In this survey paper, we discuss the classical Cassels' theorem on existence
of small-height zeros of quadratic forms over Q and its many extensions, to
different fields and rings, as well as to more general situations, such as
existence of totally isotropic small-height subspaces. We also discuss related
recent results on effective structural theorems for quadratic spaces, as well
as Cassels'-type theorems for small-height zeros of quadratic forms with
additional conditions. We conclude with a selection of open problems.Comment: 16 pages; to appear in the proceedings of the BIRS workshop on
"Diophantine methods, lattices, and arithmetic theory of quadratic forms", to
be published in the AMS Contemporary Mathematics serie
Special least squares solutions of the reduced biquaternion matrix equation with applications
This paper presents an efficient method for obtaining the least squares
Hermitian solutions of the reduced biquaternion matrix equation . The method leverages the real representation of reduced biquaternion
matrices. Furthermore, we establish the necessary and sufficient conditions for
the existence and uniqueness of the Hermitian solution, along with a general
expression for it. Notably, this approach differs from the one previously
developed by Yuan et al. , which relied on the complex representation
of reduced biquaternion matrices. In contrast, our method exclusively employs
real matrices and utilizes real arithmetic operations, resulting in enhanced
efficiency. We also apply our developed framework to find the Hermitian
solutions for the complex matrix equation , expanding its
utility in addressing inverse problems. Specifically, we investigate its
effectiveness in addressing partially described inverse eigenvalue problems.
Finally, we provide numerical examples to demonstrate the effectiveness of our
method and its superiority over the existing approach.Comment: 25 pages, 3 figure
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