Abstract: In this paper, the definitions of the quaternion measure
and the quaternion vector measure are introduced. The relation
between the quaternion measure and the complex vector measure
as well as the relation between the quaternion linear functional and
the complex linear functional are discussed respectively. By using
these relations, the necessary and sufficient condition to determine
the quaternion vector measure is given.
Abstract: In this paper, the similarity invariant and the upper
semi-continuity of spherical spectrum, and the spherical spectrum
properties for infinite direct sums of quaternionic operators are
characterized, respectively. As an application of some results
established, a concrete example about the computation of the
spherical spectrum of a compact quaternionic operator with form of
infinite direct sums of quaternionic matrices is also given.
Abstract: In this paper, we discuss some properties of left
spectrum and give the representation of linear preserver map the left
spectrum of diagonal quaternionic matrices.
Abstract: By the real representation of the quaternionic matrix,
an iterative method for quaternionic linear equations Ax = b is
proposed. Then the convergence conditions are obtained. At last, a
numerical example is given to illustrate the efficiency of this method.
Abstract: The method of gait identification based on the nearest neighbor classification technique with motion similarity assessment by the dynamic time warping is proposed. The model based kinematic motion data, represented by the joints rotations coded by Euler angles and unit quaternions is used. The different pose distance functions in Euler angles and quaternion spaces are considered. To evaluate individual features of the subsequent joints movements during gait cycle, joint selection is carried out. To examine proposed approach database containing 353 gaits of 25 humans collected in motion capture laboratory is used. The obtained results are promising. The classifications, which takes into consideration all joints has accuracy over 91%. Only analysis of movements of hip joints allows to correctly identify gaits with almost 80% precision.
Abstract: According to celebrated Hurwitz theorem, there exists
four division algebras consisting of R (real numbers), C (complex
numbers), H (quaternions) and O (octonions). Keeping in view
the utility of octonion variable we have tried to extend the three
dimensional vector analysis to seven dimensional one. Starting with
the scalar and vector product in seven dimensions, we have redefined
the gradient, divergence and curl in seven dimension. It is shown
that the identity n(n - 1)(n - 3)(n - 7) = 0 is satisfied only
for 0, 1, 3 and 7 dimensional vectors. We have tried to write all
the vector inequalities and formulas in terms of seven dimensions
and it is shown that same formulas loose their meaning in seven
dimensions due to non-associativity of octonions. The vector formulas
are retained only if we put certain restrictions on octonions and split
octonions.
Abstract: The real representation of the quaternionic matrix is
definited and studied. The relations between the positive (semi)define
quaternionic matrix and its real representation matrix are presented.
By means of the real representation, the relation between the positive
(semi)definite solutions of quaternionic matrix equations and those of
corresponding real matrix equations is established.
Abstract: Any rotation of a 3-dimensional object can be performed by three consecutive rotations over Euler angles. Intrinsic rotations produce the same result as extrinsic rotations in transformation. Euler rotations are the movement obtained by changing one of the Euler angles while leaving the other two constant. These Euler rotations are applied in a simple two-axis gimbals set mounted on an automotives. The values of Euler angles are [π/4, π/4, π/4] radians inside the angles ranges for a given coordinate system and these actual orientations can be directly measured from these gimbals set of moving automotives but it can occur the gimbals lock in application at [π/2.24, 0, 0] radians. In order to avoid gimbals lock, the values of quaternion must be [π/4.8, π/8.2, 0, π/4.8] radians. The four-gimbals set can eliminate gimbals lock.
Abstract: Let M be an almost split quaternionic manifold on
which its almost split quaternionic structure is defined by a three
dimensional subbundle V of ( T M) T (M)
*
Ôèù and
{F,G,H} be a local basis for V . Suppose that the (global)
(1, 2) tensor field defined[V ,V ]is defined by
[V,V ] = [F,F]+[G,G] + [H,H], where [,] denotes
the Nijenhuis bracket. In ref. [7], for the almost split-hypercomplex
structureH = J α,α =1,2,3, and the Obata
connection ÔêçH
vanishes if and only if H is split-hypercomplex.
In this study, we give a prof, in particular, prove that if either
M is a split quaternionic Kaehler manifold, or if M is a splitcomplex
manifold with almost split-complex structure F , then the
vanishing [V ,V ] is equivalent to that of all the Nijenhuis brackets
of {F,G,H}. It follows that the bundle V is trivial if and only if
[V ,V ] = 0 .
Abstract: In this work some characterizations of semi Riemannian curvature tensor on almost split quaternion Kaehler manifolds and some characterizations of Ricci tensor on almost split quaternion Kaehler manifolds are given.