A New Quaternion Hyper-Complex Space with Hyper Argument and Basic Functions via Quaternion Dynamic Equations

Chao Wang1, Zhien Li1, Ravi P. Agarwal2,3
1Department of Mathematics, Yunnan University, Kunming, China
2Department of Mathematics, Texas A &M University-Kingsville, Kingsville, USA
3Florida Institute of Technology, Melbourne, USA

Tóm tắt

Từ khóa


Tài liệu tham khảo

Adler, S.L.: Quaternionic quantum field theory. Commun. Math. Phys. 1046, 11–656 (1986)

Adler, S.L.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford Univ. Press, NewYork (1995)

Agarwal, R.P., Bohner, M., Wong, P.J.Y.: Sturm-Liouville eigenvalue problems on time scales. Appl. Math. Comput. 99, 153–166 (1999)

Agarwal, R.P., Meehan, M., O’Regan, D.: Fixed Point Theory and Applications. Cambridge Tracts in Mathematics, vol. 141. Cambridge University Press, Cambridge (2001)

Agarwal, R.P., Meehan, M., O’Regan, D., Peterson, A.: Dynamic equations on time scales: a survey. J. Comput. Appl. Math. 141, 1–26 (2002)

Alpay, D., Colombo, F., Sabadini, I.: Quaternionic de Branges Spaces and Characteristic Operator Function. Springer Briefs in Mathematics. Springer, Cham (2020)

Alpay, D., Colombo, F., Gantner, J., Sabadini, I.: A new resolvent equation for the S-functional calculus. J. Geom. Anal. 25, 1939–1968 (2015)

Alpay, D., Colombo, F., Sabadini, I.: Slice Hyperholomorphic Schur Analysis, Operator Theory: Advances and Applications, vol. 256. Birkhäuser/Springer, Cham (2016)

Alpay, D., Colombo, F., Kimsey, D.P.: The spectral theorem for quaternionic unbounded normal operators based on the S-spectrum. J. Math. Phys. 57,(2016). https://doi.org/10.1063/1.4940051

Altmann, S.L.: Rotations, Quaternions and Double Groups. Clarendon Press, Oxford (1986)

Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1978)

Aslaksen, H.: Quaternionic determinants. Math. Intell. 18, 57–65 (1996)

Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. Birkhăuser, Boston (2001)

Campos, J., Mawhin, J.: Periodic solutions of quaternionic-values ordinary differential equations. Ann. Mat. 185, 109–127 (2006)

Cayley, A.: On certain results relating to quaternions, Philos. Mag. 26, 141–145 (1845). Reprinted in the Collected Mathematical Papers, vol. 1, pp. 123–126. Cambridge University Press, Cambridge (1989)

Cerejeiras, P., Colombo, F., Kähler, U., Sabadini, I.: Perturbation of normal quaternionic operators. Trans. Am. Math. Soc. 372, 3257–3281 (2019)

Colombo, F., Gantner, J.: Fractional powers of quaternionic operators and Kato’s formula using slice hyperholomorphicity. Trans. Am. Math. Soc. 370, 1045–1100 (2018)

Colombo, F., Gantner, J.: Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes. Operator Theory: Advances and Applications, vol. 274. Birkhäuser/Springer, Cham (2019)

Colombo, F., Sabadini, I.: A structure formula for slice monogenic functions and some of its consequences. In: Hypercomplex Analysis, pp. 101–114. Trends in Mathematics. Birkhäuser Verlag, Basel (2009)

Colombo, F., Sabadini, I.: On some properties of the quaternionic functional calculus. J. Geom. Anal. 19, 601–627 (2009)

Colombo, F., Sabadini, I.: On the formulations of the quaternionic functional calculus. J. Geom. Phys. 60, 1490–1508 (2010)

Colombo, F., Sabadini, I.: The quaternionic evolution operator. Adv. Math. 227, 1772–1805 (2011)

Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative functional calculus. Theory and applications of slice hyperholomorphic functions, Progress in Mathematics, 289. Birkhäuser/Springer Basel AG, Basel (2011)

Colombo, F., Gantner, J., Kimsey, D.P.: Spectral Theory on the S-Spectrum for Quaternionic Operators. Operator Theory: Advances and Applications, vol. 270. Birkhauser/Springer, Cham (2018)

Colombo, F., Gantner, J., Pinton, S.: An introduction to hyperholomorphic spectral theories and fractional powers of vector operators. Adv. Appl. Clifford Algebr. 31, 45 (2021)

Conway, J.B.: A Course in Functional Analysis, 2nd edn. Springer, Berlin (1994)

Cortés, V.: A new construction of homogeneous quaternionic manifolds and related geometric structures. Mem. AMS 147, 1–63 (2000)

Cortés, V., Louis, J., Smyth, P., Triend, H.: On certain Kähler quotients of quaternionic Kähler manifolds. Commun. Math. Phys. 317, 787–816 (2013)

Danielewski, M., Sapa, L.: Foundations of quaternion quantum mechanics. J. Math. Phys. 12, 1–22 (2020)

Gantner, J.: Operator theory on one-sided quaternionic linear spaces: intrinsic S-functional calculus and spectral operators. Mem. Am. Math. Soc. 267, 1–101 (2020)

Gasull, A., Llibre, J., Zhang, X.: One-dimensional quaternion homogeneous polynomial differential equations. J. Math. Phys. 50, 082705 (2009)

Georgiev, S., Morais, J.: An introduction to the Hilger quaternion numbers. AIP Conf. Proc. 1558, 550–553 (2013). https://doi.org/10.1063/1.4825549

Gibbon, J.D.: A quaternionic structure in the three-dimensional Euler and ideal magneto-hydrodynamics equation. Physica 166, 17–28 (2002)

Gibbon, J.D., Holm, D.D., Kerr, R.M., Roulstone, I.: Quaternions and particle dynamics in the Euler fluid equations. Nonlinearity 19, 1969–1983 (2006)

Gürlebeck, K., Sprössig, W.: Quaternionic Analysis and Elliptic Boundary Value Problem. Mathematical Research, vol. 56. Akademie-Verlag, Berlin (1989)

Gürlebeck, K., Sprößig, W.: Quaternionic Analysis and Elliptic Boundary Value Problems. Birkhäuser, Basel (1990)

Gürlebeck, K., Sprössig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, New York (1997)

Gürlebeck, K., Sprößig, W.: Quaternionic Calculus for Engineers and Physicists. Wiley, Chichester (1997)

Hamilton, S.W.R.: Lectures on Quaternions, Royal Irish Academy. Hodges and Smith, Dublin (1853)

Hanson, A.J.: Visualizing Quaternions. Elsevier, San Francisco (2006)

Hilger, S.: Ein Maßettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. PhD thesis, Universităt Wŭrzburg (1988)

Hilger, S.: Smoothness of invariant manifolds. J. Funct. Anal. 106, 95–129 (1992)

Hilger, S.: Generalized theorem of Hartman–Grobman on measure chains. J. Austral. Math. Soc. Ser. A. 60, 157–191 (1996)

Hilger, S.: Differential and difference calculus-united! Nonlinear Anal. 30, 2683–2694 (1997)

Hilscher, R.: Linear Hamiltonian systems on time scales: transformations. Dyn. Syst. Appl. 8(3–4), 489–501 (1999)

Kou, K.I., Xia, Y.: Linear quaternion differential equations: basic theory and fundamental results. Stud. Appl. Math. 141, 3–45 (2018)

Kuipers, J.B.: Quaternions and Rotation Sequences—A Primer with Applications to Orbits, Aerospace, and Virtual Reality. Princeton University Press, Princeton, NJ (1999)

Lanford, O.E.: Time evolution of large classical systems. In: Moser, J. (ed.) Dynamical Systems, Theory and Applications. Lecture Notes in Physics, vol. 38, pp. 1–111. Springer, Berlin (1975)

Leo, S.D., Ducati, G.C.: Solving simple quaternionic differential equations. J. Math. Phys. 44, 2224–2233 (2003)

Li, Z., Wang, C.: Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales. Open Math. 18, 353–377 (2020)

Li, C., McIntosh, A., Qian, T.: Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces. Rev. Mat. Iberoam. 10, 665–721 (1994)

Li, Z., Wang, C., Agarwal, R.P., O’Regan, D.: Commutativity of quaternion-matrix-valued functions and quaternion matrix dynamic equations on time scales. Stud. Appl. Math. 146, 139–210 (2021)

Meister, L., Schaeben, H.: A concise quaternion geometry of rotations. Math. Meth. Appl. Sci. 28, 101–126 (2005)

Morais, J., Cação, I.: Quaternion Zernike spherical polynomials. Math. Comput. 84, 1317–1337 (2015)

Morais, J., Kou, K.I.: Constructing prolate spheroidal quaternion wave functions on the sphere. Math. Methods Appl. Sci. 39, 3961–3978 (2016)

Morais, J., Georgiev, S., Sprößig, W.: Real Quaternionic Calculus Handbook. Springer, Basel (2010)

Neumann, J.V.: Mathematical Foundations of Quantum Mechanics (Trans. R.T. Beyer). Princeton University Press, Princeton (1955)

Qian, T.: Singular integrals on the n-torus and its Lipschitz perturbations. In: Clifford Algebras in Analysis and Related Topics. Studies in Advanced Mathematics, pp. 94–108. CRC Press, Boca Raton (1996)

Qian, T.: Singular integrals on star-shaped Lipschitz surfaces in the quaternionic space. Math. Ann. 310, 601–630 (1998)

Qian, T., Ryan, J.: Conformal transformations and Hardy spaces arising in Clifford analysis. J. Oper. Theory. 35, 349–372 (1996)

Qian, T., Yang, Y.: Hilbert transforms on the sphere with the Clifford algebra setting. J. Fourier Anal. Appl. 15, 753–774 (2009)

Rodman, L.: Topic in Quaternion Linear Algebra. Princeton University Press, Princeton (2014)

Roubtsov, V.N., Roulstone, I.: Examples of quaternionic and Kähler structures in Hamiltonian models of nearly geostrophic flow. J. Phys. A. 30, 63–68 (1997)

Roubtsov, V.N., Roulstone, I.: Holomorphic structures in hydrodynamical models of nearly geostrophic flow. Proc. R. Soc. Lond. A 457, 1519–1531 (2001)

Ryan, J.: Clifford analysis with generalized elliptic and quasi-elliptic functions. Appl. Anal. 13, 151–171 (1982)

Sprößig, W.: On decompositions of the Clifford valued Hilbert space and their applications to boundary value problems. Adv. Appl. Clifford Algebr. 5, 167–186 (1995)

Sudbery, A.: Quaternionic analysis. Math. Proc. Camb. Philos. Soc. 85, 199–224 (1979)

Viswanath, K.: Normal operators on quaternionic Hilbert space. Trans. Am. Math. Soc. 162, 337–350 (1971)

Wang, C., Agarwal, R.P.: Almost periodic solution for a new type of neutral impulsive stochastic Lasota–Wazewska timescale model. Appl. Math. Lett. 70, 58–65 (2017)

Wang, C., Agarwal, R.P.: Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations. Discret. Contin. Dyn. Syst. Ser. B 25, 781–798 (2020)

Wang, C., Agarwal, R.P., Sakthivel, R.: Almost periodic oscillations for delay impulsive stochastic Nicholson’s blowflies timescale model. Comput. Appl. Math. 37, 3005–3026 (2018)

Wang, C., Agarwal, R.P., O’Regan, D.: Calculus of fuzzy vector-valued functions and almost periodic fuzzy vector-valued functions on time scales. Fuzzy. Sets Syst. 375, 1–52 (2019)

Wang, C., Agarwal, R.P., O’Regan, D., Sakthivel, R.: Theory of Translation Closedness for Time Scales, Developments in Mathematics, vol. 62. Springer, Switzerland (2020)

Wilczyński, P.: Quaternionic-valued ordinary differential equations. The Riccati equation. J. Differ. Equ. 247, 2163–2187 (2009)

Wilczyński, P.: Quaternionic-valued ordinary differential equations II. Coinciding sectors. J. Differ. Equ. 252, 4503–4528 (2012)

Xu, D., Xia, Y., Mandic, D.P.: Optimization in quaternion dynamic systems: gradient, hessian, and learning algorithms. IEEE Trans. Neural Netw. Learn. Syst. 27, 249–261 (2016)

Zhang, F.: Gersgorin type theorems for quaternionic matrices. Linear Algebr. Appl. 424, 139–153 (2007)

Zhang, X.: Global structure of quaternion polynomial differential equations. Commun. Math. Phys. 303, 301–316 (2011)

Zhang, F., Wei, Y.: Jordan canonical from of a partitioned complex matrix and its application to real quaternion matrices. Commun. Algebr. 29, 2363–2375 (2001)