Quaternion Algebra

In this paper we revisit the topic of how to formulate error terms for estimation problems that involve rotational state variables. We present a firstprinciples linearization approach that yields multiplicative error terms for unit-length quaternion representations of rotations, as well as for canonical rotation matrices. Quaternion algebra is employed throughout our derivations. We show the utility of our approach through two examples: (i) linearizing a sun sensor measurement error term, and (ii) weighted-least-squares point-cloud alignment. ; tim.barfoot@utoronto.ca parameters), the standard 3 × 3 rotation matrix, and Euler angles are all common [4].

This paper is the second one of a series of three and it is the continuation of [1]. We review some properties of the algebraic spinors in Cℓ 3,0 and Cℓ 0,3 and how Weyl, Pauli and Dirac spinors are constructed in Cℓ 3,0 (and Cℓ 0,3 , in the case of Weyl spinors). A plane wave solution for the Dirac equation is obtained, and the Dirac equation is written in terms of Weyl spinors, and alternatively, in terms of Pauli spinors. Finally the covariant and contravariant undotted spinors in Cℓ 0,3 ≃ H ⊕ H are constructed. We prove that there exists an application that maps Cℓ + 0,3 , viewed as a right H-module, onto Cℓ + 0,3 , but now viewed as a left H-module.

Currently, many six degree of freedom (6-DOF) trajectory simulations and simulations of gyroscopic motion use quaternions to define a vehicle's orientation. Of those that do, however, none take full advantage of the properties of quaternion algebra. Quaternions are also known as hypercomplex numbers. They can be treated as individual quantities for which all the standard algebraic operations are defined. Consequently, they have advantages that Euler angles and transformation matrices do not. This paper will describe the use of quaternion algebra and elliptic functions to obtain a closed form solution for torque free gyroscopic motion in terms of the rotational quaternion and its derivative. It will also define an alternative 6-DOF formulation, that, when combined with quaternion algebra, is potentially much more powerful than current simulations. Lastly, the paper will present a special case of spin stabilized rigid bodies.

This paper presents the equations for the implementation of rotational quaternions in the geometrically exact three-dimensional beam theory. A new finite-element formulation is proposed in which the rotational quaternions are used for parametrization of rotations along the length of the beam. The formulation also satisfies the consistency condition that the equilibrium and the constitutive internal force and moment vectors are equal in its weak form. A strict use of the quaternion algebra in the derivation of governing equations and for the numerical solution is presented. Several numerical examples demonstrate the validity, performance and accuracy of the proposed approach.

proposed a method that allows simultaneous computation of the rigid transformations from world frame to robot base frame and from hand frame to camera frame. Their method attempts to solve a homogeneous matrix equation of the form AX = ZB. They use quaternions to derive explicit linear solutions for X and Z. In this short paper, we present two new solutions that attempt to solve the homogeneous matrix equation mentioned above: (i) a closed-form method which uses quaternion algebra and a positive quadratic error function associated with this representation and (ii) a method based on non-linear constrained minimization and which simultaneously solves for rotations and translations. These results may be useful to other problems that can be formulated in the same mathematical form. We perform a sensitivity analysis for both our two methods and the linear method developed by Zhuang et al. 1]. This analysis allows the comparison of the three methods. In the light of this comparison the non-linear optimization method, which solves for rotations and translations simultaneously, seems to be the most stable one with respect to noise and to measurement errors.

In the literature, conventional 3D inverse dynamic models are limited in three aspects related to inverse dynamic notation, body segment parameters and kinematic formalism. First, conventional notation yields separate computations of the forces and moments with successive coordinate system transformations. Secondly, the way conventional body segment parameters are defined is based on the assumption that the inertia tensor is principal and the centre of mass is located between the proximal and distal ends. Thirdly, the conventional kinematic formalism uses Euler or Cardanic angles that are sequence-dependent and suffer from singularities. In order to overcome these limitations, this paper presents a new generic method for inverse dynamics. This generic method is based on wrench notation for inverse dynamics, a general definition of body segment parameters and quaternion algebra for the kinematic formalism.

Let f be a modular eigenform of even weight k ≥ 2 and new at a prime p dividing exactly the level with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to f a monodromy module D F M f and an L-invariant L F M f . The first goal of this paper is building a suitable p-adic integration theory that allows us to construct a new monodromy module D f and L-invariant L f , in the spirit of Darmon. We conjecture both monodromy modules are isomorphic, and in particular the two L-invariants are equal.

We give an algorithm to determine a finite set of generators of the unit group of an order in a non-split classical quaternion algebra HðKÞ over an imaginary quadratic extension K of the rationals. We then apply this method to obtain a presentation for the unit group of HðZ½ 1þ ffiffiffiffi ffi À7 p 2 Þ: As a consequence a presentation is discovered for the orthogonal group SO 3 ðZ½ 1þ ffiffiffiffi ffi À7 p 2 Þ: These results provide the first examples of a characterization of the unit group of some group rings that have an epimorphic image that is an order in a non-commutative division algebra that is not a totally definite quaternion algebra. r

Let f be a modular eigenform of even weight k>0 and new at a prime p dividing exactly the level, with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to f a monodromy module D_FM(f) and an L-invariant L_FM(f). The first goal of this paper is building a suitable p-adic integration theory that allows us

We propose QTRU, a probabilistic and multi-dimensional public key cryptosystem based on the NTRU public key cryptosystem using quaternion algebra. QTRU encrypts four data vectors in each encryption session and the only other major difference between NTRU and QTRU is that the underlying algebraic structure has been changed to a non-commutative algebraic structure. As a result, QTRU inherits the strength of NTRU and its positive points. In addition, the non-commutativity of the underlying structure of QTRU makes it much more resistant to some lattice-based attacks. After a brief description of NRTU, we begin by describing the algebraic structure used in QTRU. Further, we present the details of the key generation, encryption and decryption algorithms of QTRU and discuss the issues regarding key security, message security, and probability of successful decryption. Last but not least, QTRU's resistance against lattice-based attacks is investigated.

In this paper, we considered the theory of quasideterminants and row and column determinants. We considered the application of this theory to the solving of a system of linear equations in quaternion algebra. We established correspondence between row and column determinants and quasideterminants of matrix over quaternion algebra.

We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston's Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball goes to infinity, in fact faster than any power of the logarithm of the degree of the cover, and we give a more precise quantitative lower bound. The example manifold M is arithmetic, and the proof uses detailed number-theoretic information, at the level of the Hecke eigenvalues, to drive a geometric argument based on Fried's dynamical characterization of the fibered faces. The origin of the basic fibration M → S 1 is the modular elliptic curve E = X 0 (49), which admits multiplication by the ring of integers of Q[ √ −7]. We first base change the holomorphic differential on E to a cusp form on GL(2) over K = Q[ √ −3], and then transfer over to a quaternion algebra D/K ramified only at the primes above 7; the fundamental group of M is a quotient of the principal congruence subgroup of O * D of level 7. To analyze the topological properties of M, we use a new practical method for computing the Thurston norm, which is of independent interest. We also give a non-compact finite-volume hyperbolic 3-manifold with the same properties by using a direct topological argument. CONTENTS 1. Introduction 2. Fibered faces of the Thurston norm ball 3. The geometric theorem 4. The base manifold M 5. The automorphic structure of the cohomology of M 6. The Thurston norm and fibered faces of M 7. Proof of the main result 8. The Whitehead link complement 9. Work of Long and Reid References

We present a new classification of Clifford algebra elements. Our classification is based on the notion of quaternion type. Using this classification we develop a method for analyzing of commutators and anticommutators of Clifford algebra elements. This method allows us to find out and prove a number of new properties of Clifford algebra elements.

Is "Gravity" a deformation of "Electromagnetism"? Deformation theory suggests quantizing Special Relativity: formulate Quantum Information Dynamics SL(2,C)_h-gauge theory of dynamical lattices, with unifying gauge "group" the quantum bundle obtained from the Hopf monopole bundle underlying the quaternionic algebra and Dirac-Weyl spinors. The deformation parameter is the inverse of light speed 1/c, in duality with Planck's constant h. Then mass and electric charge form a complex coupling constant (m,q), for which the quantum determinant of the quantum group SL(2,C)_h expresses the interaction strength as a linking number 2-form. There is room for both Coulomb constant k_C and Newton's gravitational constant G_N, exponentially weaker then the reciprocal of the fine structure constant α. Thus "Gravity" emerges already "quantum", in the discrete framework of QID, based on the quantized complex harmonic oscillator: the quantized q...

In this paper, we considered the theory of quasideterminants and row and column determinants. We considered the application of this theory to the solving of a system of linear equations in quaternion algebra. We established correspondence between row and column determinants and quasideterminants of matrix over quaternion algebra.

We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole, and denoted sysπ 1. P. Buser and P. Sarnak constructed Riemann surfaces X whose systole behaves logarithmically in the genus g(X). The Fuchsian groups in their examples are principal congruence subgroups of a fixed arithmetic group with rational trace field. We generalize their construction to principal congruence subgroups of arbitrary arithmetic surfaces. The key tool is a new trace estimate valid for an arbitrary ideal in a quaternion algebra. We obtain a particularly sharp bound for a principal congruence tower of Hurwitz surfaces (PCH), namely the 4/3bound sysπ 1 (X PCH) ≥ 4 3 log(g(X PCH)). Similar results are obtained for the systole of hyperbolic 3-manifolds, relative to their simplicial volume.

We describe a collection of computer scripts written in PARI/GP to compute, for reflection groups determined by finite-volume polyhedra in H 3 , the commensurability invariants known as the invariant trace field and invariant quaternion algebra. Our scripts also allow one to determine arithmeticity of such groups and the isomorphism class of the invariant quaternion algebra by analyzing its ramification.

In this paper we further develop the method of quaternion typification of Clifford algebra elements suggested by the author in the previous paper. On the basis of new classification of Clifford algebra elements it is possible to reveal and prove a number of new properties of Clifford algebra. We use k-fold commutators and anticommutators. In this paper we consider Clifford and exterior degrees and elementary functions of Clifford algebra elements.

I consider differential of mapping f of continuous division ring as linear mapping the most close to mapping f. Different expressions which correspond to known deffinition of derivative are supplementary. I explore the Gateaux derivative of higher order and Taylor series. The Taylor series allow solving of simple differential equations. As an example of solution of differential equation I considered a model of exponent. I considered application of obtained theorems to complex field and quaternion algebra. In contrast to complex field in quaternion algebra congugation is linear function of original number a̅=a+iai+jaj+kak . In quaternion algebra this difference leads to the absence of analogue of the Cauchy Riemann equations that are well known in the theory of complex function.

In this paper, we compare three inverse kinematic formulation methods for the serial industrial robot manipulators. All formulation methods are based on screw theory. Screw theory is an effective way to establish a global description of rigid body and avoids singularities due to the use of the local coordinates. In these three formulation methods, the first one is based on quaternion algebra, the second one is based on dual-quaternions, and the last one that is called exponential mapping method is based on matrix algebra. Compared with the matrix algebra, quaternion algebra based solutions are more computationally efficient and they need less storage area. The method which is based on dual-quaternion gives the most compact and computationally efficient solution. Paden-Kahan sub-problems are used to derive inverse kinematic solutions. 6-DOF industrial robot manipulator's forward and inverse kinematic equations are derived using these formulation methods. Simulation and experimental results are given.

Single-case, longitudinal studies of the threedimensional vestibulo-ocular response (VOR) were conducted with two spaceflight subjects over a 180-day mission. For reference, a control study was performed in the laboratory with 13 healthy volunteers. Horizontal, vertical and torsional VOR was measured during active yaw, pitch and roll oscillations of the head, performed during visual fixation of real and imaginary targets. The control group was tested in the head-upright position, and in the gravity-neutral, onside and supine positions. Binocular eye movements were recorded throughout using videooculography, yielding eye position in Fick coordinates. Eye velocity was calculated using quaternion algebra. Head angular velocities were measured by a headmounted rate sensor. Eye/head velocity gain and phase were evaluated for the horizontal, vertical and torsional VOR. The inclination of Listing's plane was also calculated for each test session. Control group gain for horizontal and vertical VOR was distributed closely around unity during real-target fixation, and reduced by 30-50% during imaginary-target trials. Phase was near zero throughout. During head pitch in the onside position, vertical VOR gain did not change significantly. Analysis of up/down asymmetry indicated that vertical VOR gain for downward head movement was significantly higher than for upward head movement. Average torsional VOR gain with real-target fixation was significantly higher than with imaginary-target fixation. No difference in phase was found. In contrast to vertical VOR gain, torsional VOR gain was significantly lower in the gravityneutral supine position. Spaceflight subjects showed no notable modification of horizontal or vertical VOR gain or phase during real-target fixation over the course of the mission. However, the up/down asymmetry of vertical VOR gain was inverted in microgravity. Torsional VOR gain was clearly reduced in microgravity, with some recovery in the later phase. After landing, there was a dip in gain during the first 24 h, with subsequent recovery to near baseline over the 13-day period tested. Listing's plane appeared to remain stable throughout the mission. The findings reflect various functions of the otolith responses. The reduced torsional VOR gain in microgravity is attributed to the absence of the gravity-dependent, dynamic stimulation to the otoliths (primarily utricles). On the other hand, the reversal of vertical VOR up/down gain asymmetry in microgravity is attributed to the offloading of the constant 1-g bias (primarily to the saccules) on Earth. The observed increase in torsional VOR gain from the 1st to the 6th month in microgravity demonstrates the existence of longer-term adaptive processes than have previously been considered. Likely factors are the adaptive reweighting of neck-proprioceptive afferents and/or enhancement of efference copy.

The method for processing perturbed Keplerian systems known today as the linearization was already known in the XVIII th century; Laplace seems to be the first to have codified it. We reorganize the classical material around the Theorem of the Moving Frame. Concerning Stiefel's own contribution to the question, on the one hand, we abandon the formalism of Matrix Theory to proceed exclusively in the context of quaternion algebra; on the other hand, we explain how, in the hierarchy of hypercomplex systems, both the KS-transformation and the classical projective decomposition emanate by doubling from the Levi-Civita transformation. We propose three ways of stretching out the projective factoring into four-dimensional coordinate transformations, and offer for each of them a canonical extension into the moment space. One of them is due to Ferr~ndiz; we prove it to be none other than the extension of Burdet's focal transformation by Liouville's technique. In the course of constructing the other two, we examine the complementarity between two classical methods for transforming Hamiltonian systems, on the one hand, Stiefel's method for raising the dimensions of a system by means of weakly canonical extensions, on the other, Liouville's technique of lowering dimensions through a Reduction induced by ignoration of variables.

In [8] we constructed pairs of units u, v in Z-orders of a quaternion algebra over Q(√ −d), d ≡ 7 (mod 8) positive and square free, such that u n , v n is free for some n ∈ N. Here we extend this result to any imaginary quadratic extension of Q, thus including matrix algebras. More precisely, we show that u n , v n is a free group for all n ≥ 1 and d > 2 and for d = 2 and all n ≥ 2. The units we use arise from Pell's and Gauss' equations. A criterion for a pair of homeomorphisms to generate a free semigroup is also established and used to prove that two certain units generate a free semigroup but that, in this case, the Ping-Pong Lemma can not be applied to show that the group they generate is free.

We propose an algebraic framework for studying coherent space-time codes, based on arithmetic lattices on central simple algebras. For two transmit antennas, this algebra is called a quaternion algebra. For this reason, we call these lattices quaternionic lattices. The design criterion is the one described by V. Tarokh et al. (see IEEE Trans. Inf. Theory, vol.44, p.744-65, 1998).

Representations of two bridge knot groups in the isometry group of some complete Riemannian 3-manifolds as $E^{3}$ (Euclidean 3-space), $H^{3}$ (hyperbolic 3-space) and $ E^{2,1}$ (Minkowski 3-space), using quaternion algebra theory, are studied. We study the different representations of a 2-generator group in which the generators are send to conjugate elements, by analyzing the points of an algebraic variety, that we call the \emph{variety of affine c-representations of}$G$. Each point in this variety correspond to a representation in the unit group of a quaternion algebra and their affine deformations.

In the context of the integration over algebras introduced in a previous paper, we obtain several results for a particular class of associative algebras with identity. The algebras of this class are called self-conjugated, and they include, for instance, the paragrassmann algebras of order p, the quaternionic algebra and the toroidal algebras. We study the relation between derivations and integration, proving a generalization of the standard result for the Riemann integral about the translational invariance of the measure and the vanishing of the integral of a total derivative (for convenient boundary conditions). We consider also the possibility, given the integration over an algebra, to define from it the integral over a subalgebra, in a way similar to the usual integration over manifolds. That is projecting out the submanifold in the integration measure. We prove that this is possible for paragrassmann algebras of order p, once we consider them as subalgebras of the algebra of the (p + 1) × (p + 1) matrices. We find also that the integration over the subalgebra coincides with the integral defined in the direct way. As a by-product we can define the integration over a one-dimensional Grassmann algebra as a trace over 2 × 2 matrices.

We introduce and investigate the topological algebra of Colombeau Generalized quaternions, H. This is an important object to study if one wants to build the algebraic theory of Colombeau generalized numbers. We classify the dense ideals of K, in the algebraic sense and prove that it has a maximal ring of quotients which is Von Neumann regular. Using the classification of the dense ideals we give a criteria for a generalized holomorphic function to satisfy the identity theorem.

We present a new classification of Clifford algebra elements. Our classification is based on the notion of quaternion type. Using this classification we develop a method for analyzing commutators and anticommutators of Clifford algebra elements. This method allows us to find out and prove a number of new properties of Clifford algebra elements.

In this paper, we considered the theory of quasideterminants and row and column determinants. We considered the application of this theory to the solving of a system of linear equations in quaternion algebra. We established correspondence between row and column determinants and quasideterminants of matrix over quaternion algebra.

We give a simplified and a direct proof of a special case of Ratner's theorem on closures and uniform distribution of individual orbits of unipotent flows; namely, the case of orbits of the diagonally embedded unipotent subgroup acting on $SL(2,K)/\Gamma_1\times ...\times SL(2,K)/\Gamma_n$, where $K$ is a locally compact field of characteristic 0 and each $\Gamma_i$ is a cocompact discrete subgroup of $SL(2,K)$. This special case of Ratner's theorem plays a crucial role in the proofs of uniform distribution of Heegner points by Vatsal, and Mazur conjecture on Heegner points by C. Cornut; and their generalizations in their joint work on CM-points and quaternion algebras. A purpose of the article is to make the ergodic theoretic results accessible to a wide audience.

Consider a semigroup generated by matrices associated with an edge-coloring of a strongly connected, aperiodic digraph. We call the semigroup Lie-solvable if the Lie algebra generated by its elements is solvable. We show that if the semigroup is Lie-solvable then its kernel is a right group. Next, we study the Lie algebra generated by the kernel. Lie algebras generated by two idempotents are analyzed in detail. We find that these have homomorphic images that are generalized quaternion algebras. We show that if the kernel is not a direct product, then the Lie algebra generated by the kernel is not solvable by describing the structure of these algebras. Finally, we discuss an infinite class of examples that are shown to always produce strongly connected aperiodic digraphs having kernels that are not right groups.

We introduce a new equivalence relation for complete algebraic varieties with canonical singularities, generated by birational equivalence, by flat algebraic deformations (of varieties with canonical singularities), and by quasi-étale morphisms, i.e., morphisms which are unramified in codimension 1. We denote the above equivalence by A.Q.E.D. : = Algebraic-Quasi-Etale-Deformation. A completely similar equivalence relation, denoted by C-Q.E.D. (: = Complex-Quasi-Étale-Deformation), can be considered for compact complex manifolds and for compact complex spaces with canonical singularities. It is generated by bimeromorphic equivalence, by flat deformations of complex spaces with canonical singularities, and by quasi-étale morphisms. By a recent theorem of Siu (conjecturally holding also for compact Kähler manifolds), dimension and Kodaira dimension are invariants for A.Q.E.D. (also for C-Q.E.D.-equivalence of algebraic varieties?). It was an interesting question whether conversely two algebraic varieties of the same dimension and with the same Kodaira dimension are Q.E.D.-equivalent (the question then bifurcates into: A.Q.E.D., or at least C-Q.E.D.). The answer to the A.Q.E.D. question is positive for curves by well known results. Using Enriques' classification we show first that the answer to the C-Q.E.D. question is positive for special algebraic surfaces (i.e., for algebraic surfaces with Kodaira dimension at most 1). More generally, using Kodaira's classification we show that the same result holds for compact complex surfaces with Kodaira dimension 0, 1 and even first Betti number. The appendix by Sönke Rollenske shows that the same does not hold if we consider also surfaces with odd first Betti number. Indeed he proves that any surface which is C-Q.E.D.-equivalent to a Kodaira surface is itself a Kodaira surface. Concerning the A.Q.E.D. question, we show that the answer is positive for complex algebraic surfaces of Kodaira dimension ≤ 1. The answer to the Q.E.D. question is instead negative for surfaces of general type: the other appendix, due to Fritz Grunewald, is devoted to showing that the (rigid) Kuga-Shavel type surfaces of general type obtained as quotients of the bidisk via discrete groups constructed from quaternion algebras belong to countably many distinct Q.E.D. equivalence classes. We leave aside here the A.Q.E.D. question for algebraic surfaces of Kodaira dimension ≤ 1 defined over an algebraically closed field of positive characteristic; we hope to be able to address this question in the future. We pose then several questions, and point out possible applications of the above ideas, with the aim of showing the importance for classification theory and for other purposes of investigating Q.E.D. equivalence.

Is "Gravity" a deformation of "Electromagnetism"? G N m 2 e k C e 2 ≈ 10 −54 ↔ e −1/α ≈ 10 −59. Thus "Gravity" emerges already "quantum", in the discrete framework of QID, based on the quantized complex harmonic oscillator: the quantized qubit. All looks promising, but will the details backup this "grand design scheme"? Contents

We obtain an arithmetic expression of the Selberg zeta function for cocompact Fuchsian group defined via an indefinite division quaternion algebra over Q. As application to the prime geodesic theorem, we prove certain uniformity of the distribution.

In this study, 3-D Lattice Solid Model (LSMearth or LSM) was extended by introducing particle-scale rotation. In the new model, for each 3-D particle, we introduce six degrees of freedom: Three for translational motion, and three for orientation. Six kinds of relative motions are permitted between two neighboring particles, and six interactions are transferred, i.e., radial, two shearing forces, twisting and two bending torques. By using quaternion algebra, relative rotation between two particles is decomposed into two sequence-independent rotations such that all interactions due to the relative motions between interactive rigid bodies can be uniquely decided. After incorporating this mechanism and introducing bond breaking under torsion and bending into the LSM, several tests on 2-D and 3-D rock failure under uniaxial compression are carried out. Compared with the simulations without the single particle rotational mechanism, the new simulation results match more closely experimental results of rock fracture and hence, are encouraging. Since more parameters are introduced, an approach for choosing the new parameters is presented.

In this study, 3-D Lattice Solid Model (LSMearth or LSM) was extended by introducing particle-scale rotation. In the new model, for each 3-D particle, we introduce six degrees of freedom: Three for translational motion, and three for orientation. Six kinds of relative motions are permitted between two neighboring particles, and six interactions are transferred, i.e., radial, two shearing forces, twisting and two bending torques. By using quaternion algebra, relative rotation between two particles is decomposed into two sequence-independent rotations such that all interactions due to the relative motions between interactive rigid bodies can be uniquely decided. After incorporating this mechanism and introducing bond breaking under torsion and bending into the LSM, several tests on 2-D and 3-D rock failure under uniaxial compression are carried out. Compared with the simulations without the single particle rotational mechanism, the new simulation results match more closely experimental results of rock fracture and hence, are encouraging. Since more parameters are introduced, an approach for choosing the new parameters is presented.

A new direct relativistic four-component Kramers-restricted multiconfiguration self-consistent-field ͑KR-MCSCF͒ code for molecules has been implemented. The program is based upon Kramers-paired spinors and a full implementation of the binary double groups ͑D 2h * and subgroups͒. The underlying quaternion algebra for one-electron operators was extended to treat two-electron integrals and density matrices in an efficient and nonredundant way. The iterative procedure is direct with respect to both configurational and spinor variational parameters; this permits the use of large configuration expansions and many basis functions. The relativistic minimum-maximum principle is implemented in a second-order restricted-step optimization algorithm, which provides sharp and well-controlled convergence. This paper focuses on the necessary modifications of nonrelativistic MCSCF methodology to obtain a fully variational KR-MCSCF implementation. The general implementation also allows for the use of molecular integrals from a two-component relativistic Hamiltonian as, for example, the Douglas-Kroll-Hess variants. Several sample applications concern the determination of spectroscopic properties of heavy-element atoms and molecules, demonstrating the influence of spin-orbit coupling in MCSCF approaches to such systems and showing the potential of the new method.

The quadratic and cubic arithmetic geometric means (AGMs) are known to be parametrized by Jacobian and bidimensional theta series respectively. We suggest an approach based on codes of length L over an alphabet of size s to parametrize the general AGM of order L and parameter s. This is possible given the existence of number fields and quaternion algebras equipped with suitable quadratic forms. Expressions for theta series of lattices over the Gauss, Eisenstein and Hurwitz integers using Hamming weight enumerators of, respectively, binary, ternary and quaternary codes are derived.

We give a simplified and a direct proof of a special case of Ratner's theorem on closures and uniform distribution of individual orbits of unipotent flows; namely, the case of orbits of the diagonally embedded unipotent subgroup acting on $SL(2,K)/\Gamma_1\times ...\times SL(2,K)/\Gamma_n$, where $K$ is a locally compact field of characteristic 0 and each $\Gamma_i$ is a cocompact discrete subgroup of $SL(2,K)$. This special case of Ratner's theorem plays a crucial role in the proofs of uniform distribution of Heegner points by Vatsal, and Mazur conjecture on Heegner points by C. Cornut; and their generalizations in their joint work on CM-points and quaternion algebras. A purpose of the article is to make the ergodic theoretic results accessible to a wide audience.

In this paper, we analyze the performance of an Algebraic Space Time Codes (ASTC), called the Golden code. Due to its Algebraic construction based on Quaternionic algebra, the code has a full rate, full diversity, non-vanishing constant minimum determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping. In this paper, we analyze the performances of the Golden codes in correlated Rayleigh channel. We consider a coherent demodulator and we analyze the channel estimation error impact.

In this paper we present a singularity free trajectory tracking method for the cooperative working of multi-arm robot manipulators. It is based on an inverse kinematic transformation which determines the manipulator's joint angles corresponding to the end-effector trajectory given in the task space. The kinematic problem of multi-arm robot system is solved by using screw theory and quaternion algebra. Screw theory is an effective way to establish a global description of rigid body and avoids singularities due to the use of the local coordinates. Dualquaternion is the most compact and efficient dual operator to express screw displacement. Inverse kinematic solutions are obtained by using Paden-Kahan sub-problems. The proposed method is implemented using two Stäubli TX60 robot arms and simulation results are given.

We propose QTRU, a probabilistic and multi-dimensional public key cryptosystem based on the NTRU public key cryptosystem using quaternion algebra. QTRU encrypts four data vectors in each encryption session and the only other major difference between NTRU and QTRU is that the underlying algebraic structure has been changed to a non-commutative algebraic structure. As a result, QTRU inherits the strength of NTRU and its positive points. In addition, the non-commutativity of the underlying structure of QTRU makes it much more resistant to some lattice-based attacks. After a brief description of NRTU, we begin by describing the algebraic structure used in QTRU. Further, we present the details of the key generation, encryption and decryption algorithms of QTRU and discuss the issues regarding key security, message security, and probability of successful decryption. Last but not least, QTRU's resistance against lattice-based attacks is investigated.