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Journal articles on the topic 'Quaternionic function theory'

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Published: 10 March 2023

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1

Kim, Ji, and Kwang Shon. "Expansion of implicit mapping theory to split-quaternionic maps in Clifford analysis." Filomat 35, no. 11 (2021): 3833–40. http://dx.doi.org/10.2298/fil2111833k.

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This paper presents the regularity of a split-quaternionic function and a corresponding split- Cauchy-Riemann system of a split quaternion. The properties of an inverse and an implicit mapping theory for a split-quaternionic map are investigated. In addition, the paper proposes a definition and expression for a split biregular mapping in an open set in C2. The obtained results are illustrated with some examples.

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2

Pap, Margit, and Ferenc Schipp. "Quaternionic Blaschke Group." Mathematics 7, no. 1 (December 31, 2018): 33. http://dx.doi.org/10.3390/math7010033.

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In the complex case, the Blaschke group was introduced and studied. It turned out that in the complex case this group plays important role in the construction of analytic wavelets and multiresolution analysis in different analytic function spaces. The extension of the wavelet theory to quaternion variable function spaces would be very beneficial in the solution of many problems in physics. A first step in this direction is to give the quaternionic analogue of the Blaschke group. In this paper we introduce the quaternionic Blaschke group and we study the properties of this group and its subgroups.

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3

Gentili, Graziano, and Caterina Stoppato. "Geometric function theory over quaternionic slice domains." Journal of Mathematical Analysis and Applications 495, no. 2 (March 2021): 124780. http://dx.doi.org/10.1016/j.jmaa.2020.124780.

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4

Morais, J., and M. A. Pérez-de la Rosa. "Towards a quaternionic function theory linked with the Lamé's wave functions." Mathematical Methods in the Applied Sciences 38, no. 17 (January 19, 2015): 4365–87. http://dx.doi.org/10.1002/mma.3376.

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5

Alpay, Daniel, Fabrizio Colombo, and Irene Sabadini. "Perturbation of the generator of a quaternionic evolution operator." Analysis and Applications 13, no. 04 (April 28, 2015): 347–70. http://dx.doi.org/10.1142/s0219530514500249.

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The theory of slice hyperholomorphic functions, introduced in recent years, has important applications in operator theory. The quaternionic version of this function theory and its Cauchy formula yield to a definition of the quaternionic version of the Riesz–Dunford functional calculus which is based on the notion of S-spectrum. This quaternionic functional calculus allows to define the quaternionic evolution operator which appears in the quaternionic version of quantum mechanics proposed by J. von Neumann and later developed by S. L. Adler. Generation results such as the Hille–Phillips–Yosida theorem have been recently proved. In this paper, we study the perturbation of the generator. The motivation of this study is that, as it happens in the classical case of closed complex linear operators, to verify the generation conditions of the Hille–Phillips–Yosida theorem, in the concrete cases, is often difficult. Thus in this paper we study the generation problem from the perturbation point of view. Precisely, given a quaternionic closed operator T that generates the evolution operator [Formula: see text] we study under which condition a closed operator P is such that T + P generates the evolution operator [Formula: see text]. This paper is addressed to people working in different research areas such as hypercomplex analysis and operator theory.

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6

Konno, Norio, Kaname Matsue, Hideo Mitsuhashi, and Iwao Sato. "Quaternionic quantum walks of Szegedy type and zeta functions of graphs." Quantum Information and Computation 17, no. 15&16 (December 2017): 1349–71. http://dx.doi.org/10.26421/qic17.15-16-6.

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We define a quaternionic extension of the Szegedy walk on a graph and study its right spectral properties. The condition for the transition matrix of the quaternionic Szegedy walk on a graph to be quaternionic unitary is given. In order to derive the spectral mapping theorem for the quaternionic Szegedy walk, we derive a quaternionic extension of the determinant expression of the second weighted zeta function of a graph. Our main results determine explicitly all the right eigenvalues of the quaternionic Szegedy walk by using complex right eigenvalues of the corresponding doubly weighted matrix. We also show the way to obtain eigenvectors corresponding to right eigenvalues derived from those of doubly weighted matrix.

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7

Abreu Blaya, Ricardo, Juan Bory Reyes, Alí Guzmán Adán, and Uwe Kaehler. "On some structural sets and a quaternionic (φ,ψ)-hyperholomorphic function theory." Mathematische Nachrichten 288, no. 13 (April 14, 2015): 1451–75. http://dx.doi.org/10.1002/mana.201300072.

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8

Shpakivskyi, V. S., and T. S. Kuzmenko. "Integral theorems for the quaternionic G-monogenic mappings." Analele Universitatii "Ovidius" Constanta - Seria Matematica 24, no. 2 (June 1, 2016): 271–81. http://dx.doi.org/10.1515/auom-2016-0042.

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Abstract In the paper [1] considered a new class of quaternionic mappings, so- called G-monogenic mappings. In this paper we prove analogues of classical integral theorems of the holomorphic function theory: the Cauchy integral theorems for surface and curvilinear integrals, and the Cauchy integral formula for G-monogenic mappings.

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9

Shapiro, M. V., and N. L. Vasilevski. "Quaternionic ψ-hyperholomorphic functions, singular integral operators and boundary value problems I. ψ-hyperholomorphic function theory." Complex Variables, Theory and Application: An International Journal 27, no. 1 (January 1995): 17–46. http://dx.doi.org/10.1080/17476939508814803.

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10

Kravchenko, V. V., and M. V. Shapiro. "Helmholtz operator with a quaternionic wave number and associated function theory. II. Integral representations." Acta Applicandae Mathematicae 32, no. 3 (September 1993): 243–65. http://dx.doi.org/10.1007/bf01082451.

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11

Luna-Elizarrarás, M. E., J. Morais, M. A. Pérez-de la Rosa, and M. Shapiro. "On a version of quaternionic function theory related to Chebyshev polynomials and modified Sturm-Liouville operators." Quarterly of Applied Mathematics 74, no. 1 (December 7, 2015): 165–87. http://dx.doi.org/10.1090/qam/1412.

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12

Konno, Norio, Hideo Mitsuhashi, and Iwao Sato. "The quaternionic second weighted zeta function of a graph and the Study determinant." Linear Algebra and its Applications 510 (December 2016): 92–109. http://dx.doi.org/10.1016/j.laa.2016.08.007.

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13

GALEEVA, R., and A. VERJOVSKY. "QUATERNION DYNAMICS AND FRACTALS IN ℝ4." International Journal of Bifurcation and Chaos 09, no. 09 (September 1999): 1771–75. http://dx.doi.org/10.1142/s0218127499001255.

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In this paper we study the Fatou–Julia theory for some quaternionic rational functions in the quaternion skew-field ℍ. We obtain new dynamically-defined fractals in ℝ4 as the corresponding Julia sets. We also define the quaternionic Mandelbrot set.

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14

Kim, Ji-Eun. "Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System." Axioms 10, no. 3 (August 30, 2021): 206. http://dx.doi.org/10.3390/axioms10030206.

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The step derivative of a complex function can be defined with various methods. The step direction defines a basis that is distinct from that of a complex number; the derivative can then be treated by using Taylor series expansion in this direction. In this study, we define step derivatives based on complex numbers and quaternions that are orthogonal to the complex basis while simultaneously being distinct from it. Considering previous studies, the step derivative defined using quaternions was insufficient for applying the properties of quaternions by setting a quaternion basis distinct from the complex basis or setting the step direction to which only a part of the quaternion basis was applied. Therefore, in this study, we examine the definition of quaternions and define the step derivative in the direction of a generalized quaternion basis including a complex basis. We find that the step derivative based on the definition of a quaternion has a relative error in some domains; however, it can be used as a substitute derivative in specific domains.

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15

Ghamari, Elham, and Dan Kučerovský. "Quaternions and Functional Calculus." Symmetry 11, no. 8 (July 26, 2019): 953. http://dx.doi.org/10.3390/sym11080953.

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In this paper, we develop the notion of generalized characters and a corresponding Gelfand theory for quaternionic C * -algebras. These are C*-algebras whose structure permits an action of the quaternions. Applications are made to functional calculus, and we develop an S-functional calculus related to what we term structural regular functions.

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16

Yefremov, Alexander P. "Physical theories in hypercomplex geometric description." International Journal of Geometric Methods in Modern Physics 11, no. 06 (July 2014): 1450062. http://dx.doi.org/10.1142/s0219887814500625.

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Compact description is given of algebras of poly-numbers: quaternions, bi-quaternions, double (split-complex) and dual numbers. All units of these (and exceptional) algebras are shown to be represented by direct products of 2D vectors of a local basis defined on a fundamental surface. In this math medium a series of equalities identical or similar to known formulas of physical laws is discovered. In particular, a condition of the algebras' stability with respect to transformations of the 2D-basis turns out equivalent to the spinor (Schrödinger–Pauli and Hamilton–Jacobi) equations of mechanics. It is also demonstrated that isomorphism of SO(3, 1) and SO(3, ℂ) groups leads to formulation of a quaternion relativity theory predicting all effects of special relativity but simplifying solutions of relativistic problems in non-inertial frames. Finely it is shown that the Cauchy–Riemann type equations written for functions of quaternion variable repeat vacuum Maxwell equations of electrodynamics, while a quaternion space with non-metricity comprises main relations of Yang–Mills field theory.

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17

Howe, Sean. "The spectral p-adic Jacquet–Langlands correspondence and a question of Serre." Compositio Mathematica 158, no. 2 (February 2022): 245–86. http://dx.doi.org/10.1112/s0010437x22007308.

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We show that the completed Hecke algebra of $p$ -adic modular forms is isomorphic to the completed Hecke algebra of continuous $p$ -adic automorphic forms for the units of the quaternion algebra ramified at $p$ and $\infty$ . This gives an affirmative answer to a question posed by Serre in a 1987 letter to Tate. The proof is geometric, and lifts a mod $p$ argument due to Serre: we evaluate modular forms by identifying a quaternionic double-coset with a fiber of the Hodge–Tate period map, and extend functions off of the double-coset using fake Hasse invariants. In particular, this gives a new proof, independent of the classical Jacquet–Langlands correspondence, that Galois representations can be attached to classical and $p$ -adic quaternionic eigenforms.

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18

Bisi, Cinzia, and Caterina Stoppato. "Landau’s theorem for slice regular functions on the quaternionic unit ball." International Journal of Mathematics 28, no. 03 (March 2017): 1750017. http://dx.doi.org/10.1142/s0129167x17500173.

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During the development of the theory of slice regular functions over the real algebra of quaternions [Formula: see text] in the last decade, some natural questions arose about slice regular functions on the open unit ball [Formula: see text] in [Formula: see text]. This work establishes several new results in this context. Along with some useful estimates for slice regular self-maps of [Formula: see text] fixing the origin, it establishes two variants of the quaternionic Schwarz–Pick lemma, specialized to maps [Formula: see text] that are not injective. These results allow a full generalization to quaternions of two theorems proven by Landau for holomorphic self-maps [Formula: see text] of the complex unit disk with [Formula: see text]. Landau had computed, in terms of [Formula: see text], a radius [Formula: see text] such that [Formula: see text] is injective at least in the disk [Formula: see text] and such that the inclusion [Formula: see text] holds. The analogous result proven here for slice regular functions [Formula: see text] allows a new approach to the study of Bloch–Landau-type properties of slice regular functions [Formula: see text].

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19

Wan, Dongrui, and Qianqian Kang. "Potential theory for quaternionic plurisubharmonic functions." Michigan Mathematical Journal 66, no. 1 (March 2017): 3–20. http://dx.doi.org/10.1307/mmj/1488510022.

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20

Ghiloni, Riccardo, and Alessandro Perotti. "Lagrange polynomials over Clifford numbers." Journal of Algebra and Its Applications 14, no. 05 (March 17, 2015): 1550069. http://dx.doi.org/10.1142/s0219498815500693.

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We construct Lagrange interpolating polynomials for a set of points and values belonging to the algebra of real quaternions ℍ ≃ ℝ0,2, or to the real Clifford algebra ℝ0,3. In the quaternionic case, the approach by means of Lagrange polynomials is new, and gives a complete solution of the interpolation problem. In the case of ℝ0,3, such a problem is dealt with here for the first time. Elements of the recent theory of slice regular functions are used. Leaving apart the classical cases ℝ0,0 ≃ ℝ, ℝ0,1 ≃ ℂ and the trivial case ℝ1,0 ≃ ℝ⊕ℝ, the interpolation problem on Clifford algebras ℝp,q with (p,q) ≠ (0,2), (0,3) seems to have some intrinsic difficulties.

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21

HE, JIAN-XUN, and BO YU. "WAVELET ANALYSIS OF QUATERNION-VALUED TIME-SERIES." International Journal of Wavelets, Multiresolution and Information Processing 03, no. 02 (June 2005): 233–46. http://dx.doi.org/10.1142/s0219691305000804.

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In this paper we introduce quaternion-valued multi-resolution analysis. Applying the theory of matrix-valued wavelet analysis, we give the construction of scaling functions and wavelets by identifying the quaternion-valued function with the complex duplex matrix-valued function.

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22

Alpay, Daniel, Fabrizio Colombo, Jonathan Gantner, and David P. Kimsey. "Functions of the infinitesimal generator of a strongly continuous quaternionic group." Analysis and Applications 15, no. 02 (January 25, 2017): 279–311. http://dx.doi.org/10.1142/s021953051650007x.

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The quaternionic analogue of the Riesz–Dunford functional calculus and the theory of semigroups and groups of linear quaternionic operators have recently been introduced and studied. In this paper, we suppose that [Formula: see text] is the quaternionic infinitesimal generator of a strongly continuous group of operators [Formula: see text] and we show how we can define bounded operators [Formula: see text], where [Formula: see text] belongs to a class of functions that is larger than the one to which the quaternionic functional calculus applies, using the quaternionic Laplace–Stieltjes transform. This class includes functions that are slice regular on the [Formula: see text]-spectrum of [Formula: see text] but not necessarily at infinity. Moreover, we establish the relation between [Formula: see text] and the quaternionic functional calculus and we study the problem of finding the inverse of [Formula: see text].

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23

Xu, Dongpo, Cyrus Jahanchahi, Clive C. Took, and Danilo P. Mandic. "Enabling quaternion derivatives: the generalized HR calculus." Royal Society Open Science 2, no. 8 (August 2015): 150255. http://dx.doi.org/10.1098/rsos.150255.

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Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis.

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24

Chanyal, B. C. "A relativistic quantum theory of dyons wave propagation." Canadian Journal of Physics 95, no. 12 (December 2017): 1200–1207. http://dx.doi.org/10.1139/cjp-2017-0080.

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Beginning with the quaternionic generalization of the quantum wave equation, we construct a simple model of relativistic quantum electrodynamics for massive dyons. A new quaternionic form of unified relativistic wave equation consisting of vector and scalar functions is obtained, and also satisfy the quaternionic momentum eigenvalue equation. Keeping in mind the importance of quantum field theory, we investigate the relativistic quantum structure of electromagnetic wave propagation of dyons. The present quantum theory of electromagnetism leads to generalized Lorentz gauge conditions for the electric and magnetic charge of dyons. We also demonstrate the universal quantum wave equations for two four-potentials as well as two four-currents of dyons. The generalized continuity equations for massive dyons in case of quantum fields are expressed. Furthermore, we concluded that the quantum generalization of electromagnetic field equations of dyons can be related to analogous London field equations (i.e., current to electromagnetic fields in and around a superconductor).

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25

Gal, Sorin G. "Elements of geometric theory for functions of quaternionic variable." Advances in Applied Clifford Algebras 10, no. 1 (June 2000): 91–106. http://dx.doi.org/10.1007/bf03042011.

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26

Kim, Ji, and Kwang Shon. "Inverse mapping theory on split quaternions in Clifford analysis." Filomat 30, no. 7 (2016): 1883–90. http://dx.doi.org/10.2298/fil1607883k.

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We give a split regular function that has a split Cauchy-Riemann system in split quaternions and research properties of split regular mappings with values in S. Also, we investigate properties of an inverse mapping theory with values in split quaternions.

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27

CECOTTI, S., S. FERRARA, and L. GIRARDELLO. "GEOMETRY OF TYPE II SUPERSTRINGS AND THE MODULI OF SUPERCONFORMAL FIELD THEORIES." International Journal of Modern Physics A 04, no. 10 (June 1989): 2475–529. http://dx.doi.org/10.1142/s0217751x89000972.

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We study general properties of the low-energy effective theory for 4D type II superstrings obtained by the compactification on an abstract (2,2) superconformal system. This is the basic step towards the construction of their moduli space. We give an explicit and general algorithm to convert the effective Lagrangian for the type IIA into that of type IIB superstring defined by the same (2,2) superconformal system (and vice versa). This map converts Kahler manifolds into quaternionic ones (and quaternionic into Kahlerian ones) and has a deep geometrical meaning. The relationship with the theory of normal quaternionic manifolds (and algebras), as well as with Jordan algebras, is outlined. It turns out that only a restricted class of quarternionic geometries is allowed in the string case. We derive a general and explicit formula for the (fully nonlinear) couplings of the vector-multiplets (IIA case) in terms of the basic three-point functions of the underlying superconformal theory. A number of illustrative examples is also presented.

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28

Gentili, Graziano, and Daniele C. Struppa. "A new theory of regular functions of a quaternionic variable." Advances in Mathematics 216, no. 1 (December 2007): 279–301. http://dx.doi.org/10.1016/j.aim.2007.05.010.

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29

Kim, Henry H., and Yichao Zhang. "Divisor function for quaternion algebras and application to fourth moments of L-functions." Journal of Number Theory 129, no. 12 (December 2009): 3000–3019. http://dx.doi.org/10.1016/j.jnt.2009.05.009.

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30

GIARDINO, SERGIO, and PAULO TEOTÔNIO-SOBRINHO. "A NONASSOCIATIVE QUATERNION SCALAR FIELD THEORY." Modern Physics Letters A 28, no. 35 (October 30, 2013): 1350163. http://dx.doi.org/10.1142/s0217732313501630.

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A nonassociative Groenewold–Moyal (GM) plane is constructed using quaternion-valued function algebras. The symmetrized multiparticle states, the scalar product, the annihilation/creation algebra and the formulation in terms of a Hopf algebra are also developed. Nonassociative quantum algebras in terms of position and momentum operators are given as the simplest examples of a framework whose applications may involve string theory and nonlinear quantum field theory.

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31

Yin, Ming, Wei Liu, Jun Shui, and Jiangmin Wu. "Quaternion Wavelet Analysis and Application in Image Denoising." Mathematical Problems in Engineering 2012 (2012): 1–21. http://dx.doi.org/10.1155/2012/493976.

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The quaternion wavelet transform is a new multiscale analysis tool. Firstly, this paper studies the standard orthogonal basis of scale space and wavelet space of quaternion wavelet transform in spatialL2(R2), proves and presents quaternion wavelet’s scale basis function and wavelet basis function concepts in spatial scale spaceL2(R2;H), and studies quaternion wavelet transform structure. Finally, the quaternion wavelet transform is applied to image denoising, and generalized Gauss distribution is used to model QWT coefficients’ magnitude distribution, under the Bayesian theory framework, to recover the original coefficients from the noisy wavelet coefficients, and so as to achieve the aim of denoising. Experimental results show that our method is not only better than many of the current denoising methods in the peak signal to noise ratio (PSNR), but also obtained better visual effect.

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32

He, Jianxun, Zhiqin Lu, and Jinxia Li. "On the stability of wavelet frames of quaternionic-valued functions." International Journal of Wavelets, Multiresolution and Information Processing 18, no. 02 (November 12, 2019): 2050002. http://dx.doi.org/10.1142/s0219691320500022.

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Let [Formula: see text] be a space of square integrable quaternionic-valued functions defined on real line. In this paper, for [Formula: see text] [Formula: see text] and [Formula: see text], if the sequence of functions [Formula: see text] is a wavelet frame of [Formula: see text], we study the stability of the wavelet frame when the sampling [Formula: see text] or the mother wavelet [Formula: see text] has perturbation by using the theory of wavelet analysis.

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33

Alesker, S. "Quaternionic plurisubharmonic functions and their applications to convexity." St. Petersburg Mathematical Journal 19, no. 01 (December 12, 2007): 1–14. http://dx.doi.org/10.1090/s1061-0022-07-00982-x.

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34

Gürlebeck, K., and H. R. Malonek. "On strict inclusions of weighted dirichlet spaces of monogenic functions." Bulletin of the Australian Mathematical Society 64, no. 1 (August 2001): 33–50. http://dx.doi.org/10.1017/s0004972700019663.

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We consider a scale of weighted spaces a quaternion-valued functions of three real variables. This scale generalises the idea of Qp-spaces in complex function theory. The goal of this paper is to prove that the inclusions of spaces from the scale are strict inclusion. As a tool we prove some properties of special monogenic polynomials which have an importance in their own right independently of their use in the scale of Qp-spaces.

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35

Yang, Yan Rong. "Spatial Vector Data Indexing Network Clustered Mode Based on P2P." Applied Mechanics and Materials 513-517 (February 2014): 2339–44. http://dx.doi.org/10.4028/www.scientific.net/amm.513-517.2339.

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With the expansion of a variety of power networks, communication networks and computer network and in order to meet the needs of network users, network services must be promoted by the use of dynamic network environment. But, the workload of the server is very large. According to P2P space vector theory and space vector quaternion theory, this paper studies the network communication networks clustered mode and establishes quaternion model of P2P space vector. At last, this paper treats the index of communications network cluster as the experimental model and establishes quaternion-node network of communication network. It also stimulates quaternion model of P2P space vector and gets the server function achievement result of two-node in P2P spatial pattern which provide a theoretical reference for the innovation of communication and computer network servers.

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36

Lü, Guangshi, and Honggang Xia. "Note on divisor function for quaternion algebras." Journal of Number Theory 130, no. 10 (October 2010): 2147–56. http://dx.doi.org/10.1016/j.jnt.2010.03.008.

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37

Sands, Jonathan W. "Twisted zeta functions of quaternion orders." Journal of Number Theory 160 (March 2016): 32–43. http://dx.doi.org/10.1016/j.jnt.2015.08.006.

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38

El-Sayed Ahmed, A., and Saleh Omran. "Weighted classes of quaternion-valued functions." Banach Journal of Mathematical Analysis 6, no. 2 (2012): 180–91. http://dx.doi.org/10.15352/bjma/1342210168.

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39

Uysal, Mine, and Engin Özkan. "Higher-Order Jacobsthal–Lucas Quaternions." Axioms 11, no. 12 (November 25, 2022): 671. http://dx.doi.org/10.3390/axioms11120671.

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In this work, we define higher-order Jacobsthal–Lucas quaternions with the help of higher-order Jacobsthal–Lucas numbers. We examine some identities of higher-order Jacobsthal–Lucas quaternions. We introduce their basic definitions and properties. We give Binet’s formula, Cassini’s identity, Catalan’s identity, d’Ocagne identity, generating functions, and exponential generating functions of the higher-order Jacobsthal–Lucas quaternions. We also give some relations between higher-order Jacobsthal and Jacobsthal–Lucas quaternions.

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40

Kim, Ji Eun. "Differentiability of pseudo-dual-quaternionic functions with a differential operator." Journal of Nonlinear Sciences and Applications 11, no. 11 (August 17, 2018): 1235–42. http://dx.doi.org/10.22436/jnsa.011.11.03.

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41

Cervantes, J. Oscar González. "On Cauchy Integral Theorem for Quaternionic Slice Regular Functions." Complex Analysis and Operator Theory 13, no. 6 (March 14, 2019): 2527–39. http://dx.doi.org/10.1007/s11785-019-00913-2.

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42

PAPIKIAN, MIHRAN. "ON GENERATORS OF ARITHMETIC GROUPS OVER FUNCTION FIELDS." International Journal of Number Theory 07, no. 06 (September 2011): 1573–87. http://dx.doi.org/10.1142/s1793042111004265.

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Let F = 𝔽q(T) be the field of rational functions with 𝔽q-coefficients, and A = 𝔽q[T] be the subring of polynomials. Let D be a division quaternion algebra over F which is split at 1/T. For certain A-orders in D we find explicit finite sets generating their groups of units.

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43

Böckle, Gebhard, and Ralf Butenuth. "On computing quaternion quotient graphs for function fields." Journal de Théorie des Nombres de Bordeaux 24, no. 1 (2012): 73–99. http://dx.doi.org/10.5802/jtnb.789.

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44

Rosson, Holly J. "Theta Series of Quaternion Algebras over Function Fields." Journal of Number Theory 94, no. 1 (May 2002): 49–79. http://dx.doi.org/10.1006/jnth.2001.2723.

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Pumplün, Susanne. "Quaternion Algebras over Curves of Genus One Without Rational Points." Algebra Colloquium 12, no. 01 (March 2005): 67–92. http://dx.doi.org/10.1142/s1005386705000076.

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Quaternion algebras over curves of genus one without rational points over perfect fields of characteristic not two are classified. As a consequence, all quaternion algebras over the function field of such a curve, which are unramified everywhere, are classified.

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Liu, Jian Lan, and Li Zhu. "Research and Implementation of Virtual Campus Roaming System Interaction Function Based on OSG." Applied Mechanics and Materials 687-691 (November 2014): 2219–23. http://dx.doi.org/10.4028/www.scientific.net/amm.687-691.2219.

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The development of computer simulation technology makes virtual reality technology gradually applied to various fields. The implementation of virtual campus roaming system is an important embodiment. This paper combines with 3 d graphics rendering engine Open Scene Graph to study virtual campus interactive roaming system with a variety of functions. First of all, this paper introduces basic technologies interactive functions needed such as matrix transformation, quaternion in OSG, expounds OSG scene roaming theory, key research and implements human-computer interaction function of collision detection, scene roaming and path roaming, which introduces the switch between scene roaming, viewpoint following, path roaming by button C, and make campus roaming system have good immersion and realities and real-time performance.

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Chen, Dan, Michal Fečkan, and JinRong Wang. "Linear quaternion-valued difference equations: Representation of solutions, controllability, and observability." Journal of Mathematical Physics 63, no. 11 (November 1, 2022): 112701. http://dx.doi.org/10.1063/5.0100608.

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In this paper, we present the fundamental theory of linear quaternion-valued difference equations. Firstly, we derive general solutions for linear homogeneous equations and give the algorithm for calculating the fundamental matrix in the case of the diagonalizable form and Jordan form. Secondly, we apply the variation of the constant formula and Z transformation to study general solutions of linear nonhomogeneous equations. We obtain the representation of solutions in the case of quaternion and complex numbers. Thirdly, we adopt the ideas from the Gram matrix and the rank of the criteria to establish sufficient and necessary conditions to guarantee that linear quaternion-valued difference equations are controllable and observable in the sense of quaternion-valued and complex numbers, respectively. In addition, a direct method to solve the control function and duality is also given. Finally, we illustrate our theoretical results with some examples.

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Li, Jing. "Dual Number-Based Relative Position and Attitude Coordinated Control for Multi-Rigid-Body System." Mathematical Problems in Engineering 2018 (November 25, 2018): 1–11. http://dx.doi.org/10.1155/2018/1934174.

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The paper focuses on finding a dual number solution to position and attitude coordinated control for a multi-rigid-body system. First, a relative motion coupling model of a multi-rigid-body system is established under the framework of dual number and dual quaternion theory. Then, a coordinated control strategy that uses graph theory based on a derived new type of dual quaternion is proposed to simultaneously control the position and attitude of a multi-rigid-body system. Finally, the resulting Lyapunov function is proved to be almost globally asymptotically stable. The simulation results show that the proposed algorithm not only achieves unified control of position and attitude but also exhibits better tracking control performance.

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Virdol, Cristian. "Tate classes and poles of L-functions of twisted quaternionic Shimura surfaces." Journal of Number Theory 123, no. 2 (April 2007): 315–28. http://dx.doi.org/10.1016/j.jnt.2006.07.003.

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Luna-Elizarrarás, M. E., M. Shapiro, and V. Shpakivskyi. "On the Hausdorff Analyticity for Quaternion-Valued Functions." Complex Analysis and Operator Theory 13, no. 6 (October 20, 2018): 2863–80. http://dx.doi.org/10.1007/s11785-018-0856-8.

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