Authors:Selahattin Aslan; Yusuf Yaylı Pages: 2921 - 2931 Abstract: The shape operator is one of the most important research tools in differential geometry of surfaces. It uses the tangent vectors on the surface, extensively the tangent vectors of the parameter curves on the surface, and the normal vectors of the surfaces. Quaternions have the practical method in the rotation in the Euclidean 3-space. The main goal of our paper is to use quaternions in the research of the surfaces. Firstly, we have given the shape operator with Darboux frame along the curve in the surface. Then, the quaternionic shape operator is defined by the quaternion. Also, we have obtained some results related to the quaternionic shape operator and its matrix representation. PubDate: 2017-12-01 DOI: 10.1007/s00006-017-0804-0 Issue No:Vol. 27, No. 4 (2017)

Authors:Briceyda B. Delgado; R. Michael Porter Pages: 3015 - 3037 Abstract: We consider the inhomogeneous div-curl system (i.e. to find a vector field with prescribed div and curl) in a bounded star-shaped domain in \(\mathbb {R}^3\) . An explicit general solution is given in terms of classical integral operators, completing previously known results obtained under restrictive conditions. This solution allows us to solve questions related to the quaternionic main Vekua equation \(DW=(Df/f)\overline{W}\) in \(\mathbb {R}^3\) , such as finding the vector part when the scalar part is known. In addition, using the general solution to the div-curl system and the known existence of the solution of the inhomogeneous conductivity equation, we prove the existence of solutions of the inhomogeneous double curl equation, and give an explicit solution for the case of static Maxwell’s equations with only variable permeability. PubDate: 2017-12-01 DOI: 10.1007/s00006-017-0805-z Issue No:Vol. 27, No. 4 (2017)

Authors:Juan Du; Ron Goldman; Stephen Mann Pages: 3039 - 3062 Abstract: We flesh out the affine geometry of \({{\mathbb {R}}^3}\) represented inside the Clifford algebra \({\mathbb {R}}(4,4)\) . We show how lines and planes as well as conic sections and quadric surfaces are represented in this model. We also investigate duality between different representations of points, lines, and planes, and we show how to represent intersections between these geometric elements. Formulas for lengths, areas, and volumes are also provided. PubDate: 2017-12-01 DOI: 10.1007/s00006-017-0798-7 Issue No:Vol. 27, No. 4 (2017)

Authors:Baoling Guan; Liangyun Chen; Bing Sun Pages: 3063 - 3082 Abstract: The purpose of this paper is to study the relationships between a Hom-Lie superalgebra and its induced 3-ary-Hom-Lie superalgebra. We provide an overview of the theory and explore the structure properties such as ideals, center, derived series, solvability, nilpotency, central extensions, and the cohomology. PubDate: 2017-12-01 DOI: 10.1007/s00006-017-0801-3 Issue No:Vol. 27, No. 4 (2017)

Authors:Yueming Lu Pages: 3167 - 3181 Abstract: In this paper, we consider the Dirac-harmonic equations for differential forms with Dirichlet boundary data in weak sense. Using the Hodge-decomposition theory of differential forms and Browder–Minty theory of monotone operators, the existence is obtained. According to the Poincaré-type inequality of Dirac–Sobolev space, we get the higher integrability. PubDate: 2017-12-01 DOI: 10.1007/s00006-017-0802-2 Issue No:Vol. 27, No. 4 (2017)

Authors:Abdur Rehman; Qing-Wen Wang; Ilyas Ali; Muhammad Akram; M. O. Ahmad Pages: 3183 - 3196 Abstract: By keeping in mind the great number of applications of generalized Sylvester matrix equations in systems and control theory, in this paper we establish some necessary and sufficient conditions for the solvability to a system of eight generalized Sylvester matrix equations over the quaternion algebra. The general solution to this system is also constructed when it is solvable. Moreover, an algorithm and a numerical example are also given to make the results of this paper more practical in various fields of engineering. The findings of this paper generalize previous results in the literature. PubDate: 2017-12-01 DOI: 10.1007/s00006-017-0803-1 Issue No:Vol. 27, No. 4 (2017)

Authors:Armando Reyes; Héctor Suárez Pages: 3197 - 3224 Abstract: The aim of this paper is to investigate a general notion of \(\sigma \) -PBW extensions over Armendariz rings. As an application, the properties of Baer, quasi-Baer, p.p., and p.q.-Baer are established for these extensions. We generalize several results in the literature for Ore extensions of injective type and skew PBW extensions. PubDate: 2017-12-01 DOI: 10.1007/s00006-017-0800-4 Issue No:Vol. 27, No. 4 (2017)

Authors:Abhijit Banerjee Pages: 2321 - 2332 Abstract: We investigate the complete analytical solutions of quantum mechanical harmonic and isotonic oscillators formulated in the commutative ring of bicomplex numbers. We obtain the explicit closed form expressions for the excited eigenstates and corresponding energy eigenvalues associated with the harmonic and isotonic oscillator potentials by extending the formalism adopted in Banerjee (Ann Phys 377:493, 2017) recently to find the analytical closed form solutions for ground states. PubDate: 2017-09-01 DOI: 10.1007/s00006-017-0772-4 Issue No:Vol. 27, No. 3 (2017)

Authors:Luca Fabbri Abstract: In this paper, we consider a general twisted-curved space-time hosting Dirac spinors and we take into account the Lorentz covariant polar decomposition of the Dirac spinor field: the corresponding decomposition of the Dirac spinor field equation leads to a set of field equations that are real and where spinorial components have disappeared while still maintaining Lorentz covariance. We will see that the Dirac spinor will contain two real scalar degrees of freedom, the module and the so-called Yvon–Takabayashi angle, and we will display their field equations. This will permit us to study the coupling of curvature and torsion respectively to the module and the YT angle. PubDate: 2017-10-16 DOI: 10.1007/s00006-017-0816-9

Authors:Dong Cheng; Kit Ian Kou Abstract: The Whittaker–Shannon–Kotel’nikov (WSK) sampling theorem provides a reconstruction formula for the bandlimited signals. In this paper, a novel kind of the WSK sampling theorem is established by using the theory of quaternion reproducing kernel Hilbert spaces. This generalization is employed to obtain the novel sampling formulas for the bandlimited quaternion-valued signals. A special case of our result is to show that the 2D generalized prolate spheroidal wave signals obtained by Slepian can be used to achieve a sampling series of cube-bandlimited signals. The solutions of energy concentration problems in quaternion Fourier transform are also investigated. PubDate: 2017-10-11 DOI: 10.1007/s00006-017-0815-x

Authors:I. G. Korepanov Abstract: This is the second in a series of papers where we construct an invariant of a four-dimensional piecewise linear manifold M with a given middle cohomology class \(h\in H^{2}(M,\mathbb C)\) . This invariant is the square root of the torsion of unusual chain complex introduced in Part I of our work, multiplied by a correcting factor. Here we find this factor by studying the behavior of our construction under all four-dimensional Pachner moves, and show that it can be represented in a multiplicative form: a product of same-type multipliers over all 2-faces, multiplied by a product of same-type multipliers over all pentachora. PubDate: 2017-10-03 DOI: 10.1007/s00006-017-0811-1

Authors:Brian Jonathan Wolk Abstract: The \(SU(3)_{c}\) gauge theory for eight massless vector gauge fields is assimilated into the formalism which generated the intrinsic local gauge invariant U(1) and \(SU(2)_{L}\) theories [30, 31], thus establishing the theoretical structure for intrinsically local gauge invariant quantum chromodynamics. Use is made of both the standard and split algebra of the octonions, the last of the Hurwitz algebras, in devising the technology. Numerous interesting results are obtained, including an explanation of the inherently non-chiral nature of the strong and electromagnetic interactions in contradistinction to the weak interaction. The formalism’s novelty and universality compels contemplation of its potential ability to assimilate the gravitational interaction as well. PubDate: 2017-09-22 DOI: 10.1007/s00006-017-0812-0

Authors:Mehmet Ali Güngör; Ayşe Zeynep Azak Abstract: In this study, we define the dual complex Fibonacci and Lucas numbers. We give the generating functions and Binet formulas for these numbers. Moreover, the well-known properties e.g. Cassini and Catalan identities have been obtained for these numbers. PubDate: 2017-09-15 DOI: 10.1007/s00006-017-0813-z

Authors:Ioannis Chrysikos Abstract: Consider a Riemannian spin manifold \((M^{n}, g)\) \((n\ge 3)\) endowed with a non-trivial 3-form \(T\in \Lambda ^{3}T^{*}M\) , such that \(\nabla ^{c}T=0\) , where \(\nabla ^{c}:=\nabla ^{g}+\frac{1}{2}T\) is the metric connection with skew-torsion T. In this note we introduce a generalized \(\frac{1}{2}\) -Ricci type formula for the spinorial action of the Ricci endomorphism \({{\mathrm{Ric}}}^{s}(X)\) , induced by the one-parameter family of metric connections \(\nabla ^{s}:=\nabla ^{g}+2sT\) . This new identity extends a result described by Th. Friedrich and E. C. Kim, about the action of the Riemannian Ricci endomorphism on spinor fields, and allows us to present a series of applications. For example, we describe a new alternative proof of the generalized Schrödinger–Lichnerowicz formula related to the square of the Dirac operator \(D^{s}\) , induced by \(\nabla ^{s}\) , under the condition \(\nabla ^{c}T=0\) . In the same case, we provide integrability conditions for \(\nabla ^{s}\) -parallel spinors, \(\nabla ^{c}\) -parallel spinors and twistor spinors with torsion. We illustrate our conclusions for some non-integrable structures satisfying our assumptions, e.g. Sasakian manifolds, nearly Kähler manifolds and nearly parallel \(\hbox {G}_2\) -manifolds, in dimensions 5, 6 and 7, respectively. PubDate: 2017-09-12 DOI: 10.1007/s00006-017-0810-2

Authors:Jianquan Liao; Xingmin Li Abstract: A theory of k-analytic functions on octonions is established. The Cauchy integral formulas, Taylor series and Laurent series for the k-analytic functions are given. Moreover, we obtain the orthogonality relations for the basis of k-analytic functions. PubDate: 2017-09-08 DOI: 10.1007/s00006-017-0807-x

Authors:Pei Dang; Tao Qian; Qiuhui Chen Abstract: This paper is devoted to studying uncertainty principle of Heisenberg type for signals on the unit sphere in the Clifford algebra setting. In the Clifford algebra setting we propose two forms of uncertainty principle for spherical signals, of which both correspond to the strongest form of uncertainty principle for periodic signals. The lower-bounds of the proven uncertainty principles are in terms of a scalar-valued phase derivative. PubDate: 2017-09-06 DOI: 10.1007/s00006-017-0808-9

Authors:Juan Bory Reyes; Ricardo Abreu Blaya; Marco Antonio Pérez-de la Rosa; Baruch Schneider Abstract: The Cimmino system offers a natural and elegant generalization to four-dimensional case of the Cauchy–Riemann system of first order complex partial differential equations. Recently, it has been proved that many facts from the holomorphic function theory have their extensions onto the Cimmino system theory. In the present work a Poincaré–Bertrand formula related to the Cauchy–Cimmino singular integrals over piecewise Lyapunov surfaces in \(\mathbb {R}^4\) is derived with recourse to arguments involving quaternionic analysis. Furthermore, this paper obtains some analogues of the Hilbert formulas on the unit 3-sphere and on the 3-dimensional space for the theory of Cimmino system. PubDate: 2017-09-06 DOI: 10.1007/s00006-017-0809-8

Authors:Shi-Fang Yuan; Qing-Wen Wang; Yi-Bin Yu; Yong Tian Abstract: In this paper, we discuss Hermitian solutions of split quaternion matrix equation \(AXB+CXD=E,\) where X is an unknown split quaternion Hermitian matrix, and A, B, C, D, E are known split quaternion matrices with suitable size. The objective of this paper is to establish a necessary and sufficient condition for the existence of a solution and a solution formulas. Moreover, we provide numerical algorithms and numerical examples to exemplify the results. PubDate: 2017-09-02 DOI: 10.1007/s00006-017-0806-y