Abstract: Computer vision and robotics suffer from not having good tools for manipulating three-dimensional objects. Vectors, coordinate geometry, and trigonometry all have deficiencies. Quaternions can be used to solve many of these problems. Many properties of quaternions that are relevant to computer vision and robotics are developed. Examples are given showing how quaternions can be used to simplify derivations in computer vision and robotics.

Abstract: In this letter we show that the linear equations for the gravitational field with Heavisidian monopoles could be expressed in quaternion form using a quaternion conjugate operator as well as quaternion differential operator.

Abstract: We define indefinite quaternion Kaehlerian manifolds proving that they are Einstein if its real dimension is ≧ 8and study some conditions for the constancy of the quaternionic sectional curvature and a special behaviour of it.

Abstract: In this paper a theorem about the relation between the divisibility of orders of obstructions to cross sectioning symplectic Stiefel manifolds and quaternionic James numbers is proved. As an application of this, the existence problem of almost-quaternion k-substructures on the sphere S' is solved for all n and k except for the case n = 4m 3, k = m I for some m ? 1.

Abstract: The first examples of division algebras that were found after the quaternions belong to the class of cyclic division algebras. This class still plays a major role in the theory of central simple algebras. If is a local field, an algebraic number field, or more generally a global field, then every central division algebra over F is cyclic. This fact will be proved later; it is one of the most profound results in this book.

Abstract: The purpose of this work is to rewrite the equations of motion such that even third-class trajectories can be optimized with the current parameter optimization methods. At first the commonly used coordinate systems and Euler angles are presented in Section II. It will be realized that the definition of the Euler angles introduces additional singularities. A short derivation of the commonly-used equations of motion follows for comparison and better understanding of the later derived sets of equations of motion. Section II closes with a reduction of the optimal control problem to a parameter optimization problem. Some characteristic properties and assumptions of the parameter optimization problem are pointed out along with the necessary equations and conditions needed to solve it. Section III introduces several methods that allow integration of second- and third-class trajectories as long as some restrictions are imposed on the allowable trajectories. The first method is the so-called inertial-acceleration method. It is based on the idea that the velocity yaw angle and the velocity pitch angle can be replaced by the velocity components as measured in an inertial reference frame. The so-called two-system method is derived next. It employes the idea of having two sets of equations of motion derived in different reference frames, and thus, having their singularities at different points. In detailed discussions the problems that appear with both methods are explained, and solutions are presented, the emphasis always being on the use of these equations with optimization methods. Section III also includes a method that allows integration of third-class trajectories as long as they can be flown in the vertical plane. This method results directly from the commonly-used equations of motion after removing a restriction on the flight path angle. Because all methods of Section III have still the bank angle as the control, they are referred to here as Euler-angle methods. Section IV presents the quaternion method. Although this method has been investigated first, it is presented last because it yields the best overall solution and because many details and improvements were not found until the other methods were analyzed. Understanding the Euler-angle methods will also help in understanding the properties of the quaternion method. Because the available literature on quaternions is either complex or erroneous, the quaternion is covered in much detail. The concept of the quaternion is explained, and the rules of quaternion algebra are stated in the first two sections. Next, some necessary relationships are developed. It will then be rather straightforward to derive the actual equations of motion. How to use the quaternion method for parameter optimization methods is emphasized in the following sections

Abstract: One presents information from the arithmetic of second-order matrices used at the application of the discrete ergodic method to indefinite ternary quadratic forms (see U. M. Pachev's paper in this issue, pp. 51–84). In particular, one has developed the theory of the rotations of vector-matrices, similar to the well-known theory of B. A. Venkov for the arithmetic of the Hamiltonian quaternions.

Abstract: and the related freedom of the algebraist. The history of algebra can facilitate students' adjustment to the spirit and direction of not only abstract algebra but contemporary mathematics in general. The perplexed student might find comfort, for example, in the realization that arbitrariness and freedom are youthful additions to mathematics, born of the early nineteenth-century research on abstract algebra and non-Euclidean geometries, and widely accepted only later in the century. Around 1800 there was but one algebra. In this algebra, frequently called universal arithmetic, letters stood for numbers or quantities, and the laws of arithmetic, such as commutativity of addition and multiplication, prevailed. By the middle of the nineteenth century, however, mathematicians had created many different algebras, including the quaternions, whose multiplication is noncommutative. From this perspective, the confused modem algebra student may be a victim of the circumstance of having been born in the twentieth century. Additional solace might be derived from the revelation that bewilderment and occasionally even rejection greeted the original formulation of the symbolical approach to algebra. Thus the perplexed student has historical roots! A study of early objections to symbolical algebra can also shed light on and stimulate discussion of the oft-unvoiced concerns of present-day modern algebra students. This paper offers a newly discovered nineteenth-century manuscript which documents confu- sion as an almost immediate response to the emergence of symbolical algebra in its limited form. (Like modern algebra, early nineteenth-century symbolical algebra admitted undefined entities; unlike modern algebra, symbolical algebra, in its original form, was governed by the laws of arithmetic.) The manuscript is a spoof-play on Augustus De Morgan's Elements of Algebra (5), one of the first undergraduate algebra textbooks to incorporate the symbolical (or limited modem) approach to algebra.

Abstract: In this article the problem of computing a set of bilinear forms is considered. Our goal is to find practical algorithms for a production of two 2 × 2 matrices over a real fields complex numbers, quaternions, evaluation of convolution which requires nearly twice as smaller productions as a known algorithms.

Abstract: In this paper, we shall give explicit Z-basis of certain maximal orders of definite quaternion algebras over the rational number field Q (See Theorems below). We shall also give some remarks on symmetric maximal orders in Ponomarev [9] and Hashimoto [6] (Proposition 4.3). More precise contents are as follows. Let D be a division quaternion algebra over Q.