Skip to main content
SpringerLink
Log in

Special Numbers, Special Quaternions and Special Symbol Elements

  • Chapter
  • First Online:
Models and Theories in Social Systems

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 179))

Abstract

Most mathematical notions have connections with real life. Although the theory of rings and algebras is abstract, however, this theory has many applications, some indirect, in real life. Many sets of real-life objects, taken together with one or more laws of composition, form algebraic structures with interesting properties. Quaternion algebras and of symbol algebras have applications in various branches of mathematics, but also in computer science, physics, signal theory. In this paper we define and we study properties of \(\left( l,1,p+2q,q\cdot l\right) -\) numbers, \(\left( l,1,p+2q,q\cdot l\right) -\) quaternions, \(\left( l,1,p+2q,q\cdot l\right) -\) symbol elements. Finally, we obtain an algebraic structure with these elements.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Akyigit, M., Kosal, H.H., Tosun, M.: Fibonacci generalized quaternions. Adv. Appl. Clifford Algebras 24(3), 631–641 (2014)

    Article  MathSciNet  Google Scholar 

  • Alexandru, V., Gosoniu, N.M.: Elements of Number Theory (in Romanian). Bucharest University, Romania (1999)

    MATH  Google Scholar 

  • Alsina, M., Bayer, P.: Quaternion Orders, Quadratic Forms and Shimura Curves. CRM Monograph Series, vol. 22. American Mathematical Society, Providence (2004)

    Google Scholar 

  • Catarino, P.: A note on \(h\left(x\right)\) Fibonacci quaternion polynomials. Chaos, Solitons Fractals 77, 1–5 (2015)

    Article  MathSciNet  Google Scholar 

  • Catarino, P., Morgado, M.L.: On generalized Jacobsthal and Jacobsthal-Lucas polynomials. An. St. Univ. Ovidius Constanta Mat. Ser. 24(3), 61–78 (2016)

    Article  MathSciNet  Google Scholar 

  • Catarino, P.: The modified Pell and the modified K-Pell quaternions and octonions. Adv. Appl. Clifford Algebra 26(2), 577–590 (2016)

    Article  MathSciNet  Google Scholar 

  • Cucurezeanu, I.: Equations in Integer Numbers (in Romanian). Aramis, Romania (2006)

    Google Scholar 

  • Flatley, R.: Trace forms of symbol algebras. Algebra Colloq. 19, 1117–1124 (2012)

    Article  MathSciNet  Google Scholar 

  • Flaut, C., Savin, D., Iorgulescu, G.: Some properties of Fibonacci and Lucas symbol elements. J. Math. Sci. Adv. Appl. 20, 37–43 (2013)

    Google Scholar 

  • Flaut, C., Savin, D.: Some properties of symbol algebras of degree 3. Math. Rep. 16(66)(3), 443–463 (2014a)

    Google Scholar 

  • Flaut, C., Savin, D.: About quaternion algebras and symbol algebras. Bull. Univ. Transilv. Brasov Seria III 7(56)(2), 59–64 (2014b)

    Google Scholar 

  • Flaut, C., Savin, D.: Quaternion algebras and generalized Fibonacci-Lucas quaternions. Adv. Appl. Clifford Algebra 25(4), 853–862 (2015a)

    Article  MathSciNet  Google Scholar 

  • Flaut, C., Savin, D.: Some examples of division symbol algebras of degree 3 and 5. Carpathian J. Math. 31(2), 197–204 (2015b)

    MathSciNet  MATH  Google Scholar 

  • Flaut C., Savin D.: Some remarks regarding \(a, b, x_{0}, x_{1}\) numbers and \(a, b, x_{0}, x_{1}\) quaternions, submitted (2017)

    Google Scholar 

  • Flaut, C., Savin, D.: Some special number sequences obtained from a difference equation of degree three. Chaos, Solitons Fractals 106, 67–71 (2018)

    Article  MathSciNet  Google Scholar 

  • Flaut, C., Shpakivskyi, V.: On generalized Fibonacci quaternions and Fibonacci-Narayana quaternions. Adv. Appl. Clifford Algebra 23(3), 673–688 (2013a)

    Article  MathSciNet  Google Scholar 

  • Flaut, C., Shpakivskyi, V.: Real matrix representations for the complex quaternions. Adv. Appl. Clifford Algebra 23(3), 657–671 (2013b)

    Article  MathSciNet  Google Scholar 

  • Gille, P., Szamuely, T.: Central Simple Algebras and Galois Cohomology. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  • Halici, S.: On Fibonacci quaternions. Adv. Appl. Clifford Algebras 22(2), 321–327 (2012)

    Article  MathSciNet  Google Scholar 

  • Horadam, A.F.: Complex Fibonacci numbers and Fibonacci quaternions. Am. Math. Mon. 70, 289–291 (1963)

    Article  MathSciNet  Google Scholar 

  • Jafari, M., Yayli, Y.: Rotation in four dimensions via generalized Hamilton operators. Kuwait J. Sci. 40(1), 67–79 (2013)

    MathSciNet  Google Scholar 

  • Karatas, A., Halici, S.: Horadam octonions. An. St. Univ. Ovidius Constanta Mat. Ser. 25(3), 97–106 (2017)

    Article  MathSciNet  Google Scholar 

  • Lam, T.Y.: Introduction to Quadratic Forms over Fields. AMS, Providence (2004)

    Book  Google Scholar 

  • Ledet, A.: Brauer Type Embedding Problems. American Mathematical Society, Providence (2005)

    Book  Google Scholar 

  • Linowitz, B.: Selectivity in quaternion algebras. J. Number Theory 132, 1425–1437 (2012)

    Article  MathSciNet  Google Scholar 

  • Milne J.S.: Class Field Theory. https://www.jmilne.org/math/CourseNotes/CFT310.pdf

  • Milnor, J.: Introduction to Algebraic K-Theory. Annals of Mathematics Studies. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

  • Pierce, R.S.: Associative Algebras. Springer, Berlin (1982)

    Book  Google Scholar 

  • Ramirez, J.L.: Some combinatorial properties of the \(k\)-Fibonacci and the \(k\) -Lucas quaternions. An. St. Univ. Ovidius Constanta Mat. Ser. 23(2), 201–212 (2015)

    Google Scholar 

  • Saeid, A.B., Flaut, C., Hoskova-Mayerova, S., Afshar, M., Rafsanjani, M.K.: Some connections between BCKalgebras and nary block codes. Soft Comput. Fusion Found. Methodol. Appl. 22(1), 41–46 (2018)

    MATH  Google Scholar 

  • Savin, D., Flaut, C., Ciobanu, C.: Some properties of the symbol algebras. Carpathian J. Math. 25(2), 239–245 (2009)

    MathSciNet  MATH  Google Scholar 

  • Savin, D.: Fibonacci primes of special forms. Notes Number Theory Discret. Math. 20(2), 10–19 (2014a)

    MATH  Google Scholar 

  • Savin, D.: About some split central simple algebras. An. St. Univ. Ovidius Constanta Mat. Ser. 22(1), 263–272 (2014b)

    Article  MathSciNet  Google Scholar 

  • Savin, D.: About division quaternion algebras and division symbol algebras. Carpathian J. Math. 32(2), 233–240 (2016a)

    MathSciNet  MATH  Google Scholar 

  • Savin, D.: Quaternion algebras and symbol algebras over algebraic number field \(K,\) with the degree \(\left[K:\mathbb{Q}\right]\) even. Gulf J. Math. 4(4), 16–21 (2016b)

    MathSciNet  MATH  Google Scholar 

  • Savin, D.: About special elements in quaternion algebras over finite fields. Adv. Appl. Clifford Algebras 27(2), 1801–1813 (2017a)

    Article  MathSciNet  Google Scholar 

  • Savin D.: About split quaternion algebras over quadratic fields and symbol algebras of degree n. Bull. Math. Soc. Sci. Math. Roum. Tome 60(108)(3), 307–312 (2017b)

    Google Scholar 

  • Swamy, M.N.S.: On generalized Fibonacci quaternions. Fibonacci Quaterly 11(5), 547–549 (1973)

    Google Scholar 

  • Tarnauceanu, M.: A characterization of the quaternion group. An. St. Univ. Ovidius Constanta 21(1), 209–214 (2013)

    MathSciNet  Google Scholar 

  • Vigneras, M.F.: Arithmetique des Algebres de Quaternions. Lecture Notes in Mathematics, vol. 800. Springer, Berlin (1980)

    Chapter  Google Scholar 

  • Voight J.: The arithmetic of quaternion algebras (2010). http://www.math.dartmouth.edu/jvoight/crmquat/book/quat-modforms-041310.pdf

  • Yilmaz, N., Yazlik, Y., Taskara, N.: On the bi-periodic Lucas octonions. Adv. Appl. Clifford Algebras 27(2), 1927–1937 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author dedicates this book chapter to her mother, Prof. Elena Savin.

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Ovidius University, Bd. Mamaia 124, 900527, Constanta, Romania

    Diana Savin

Authors
  1. Diana Savin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Diana Savin .

Editor information

Editors and Affiliations

  1. Faculty of Mathematics and Computer Science, Ovidius University of Constanţa, Constanţa, Romania

    Cristina Flaut

  2. Department of Mathematics and Physics, Faculty of Military Technology, University of Defence, Brno, Czech Republic

    Šárka Hošková-Mayerová

  3. Faculty of History and Political Science, Ovidius University of Constanţa, Constanţa, Romania

    Daniel Flaut

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Savin, D. (2019). Special Numbers, Special Quaternions and Special Symbol Elements. In: Flaut, C., Hošková-Mayerová, Š., Flaut, D. (eds) Models and Theories in Social Systems. Studies in Systems, Decision and Control, vol 179. Springer, Cham. https://doi.org/10.1007/978-3-030-00084-4_23

Download citation

Publish with us

Policies and ethics

Search

Navigation