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Year 2021, Volume: 4 Issue: 3, 115 - 129, 30.09.2021

Merve Bilgin Soley Ersoy

https://doi.org/10.33434/cams.985772

Abstract

References

  • [1] A.A. Pogorui, R.M. Rodriguez-Dagnino, R.D. Rodrigue-Said, On the set of zeros of bihyperbolic polynomials, Complex Var. Elliptic Equ., 53 (2008), no. 7, 685–690.
  • [2] A. Grothendieck, Topological vector spaces, Gordon and Breach Science Publishers, New York, 1973.
  • [3] C. Segre, Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici (The real representation of complex elements and hyperalgebraic entities), Math. Annalen, 40 (1892), no. 3, 413–467.
  • [4] D. Alfsmann, H.G. Gockler, Hypercomplex bark-scale filter bank design based on allpass-phase specifications, Conference paper: Signal processing conference (EUSIPCO), Proceedings of the 20th European, Bucharest, Romania, 2012.
  • [5] D. Alpay, M.E. Luna Elizarraras, M. Shapiro, D.C. Struppa, Basics of functional analysis with bicomplex scalars and bicomplex Schur analysis, Springer Briefs in Mathematics, 2014.
  • [6] F. Catoni, D. Boccaletti, R. Cannata, ,V. Catoni, E. Nichelatti, P. Zampetti, The mathematics of Minkowski Space-Time with an introduction to commutative hypercomplex numbers, Birkhauser Verlag, Basel, Boston, Berlin, 2008.
  • [7] G. Baley Price, An introduction to multicomplex spaces and functions, Marcel Dekker Inc., New York, 1991.
  • [8] D. Bro ́d, A. Szynal-Liana, I. Włoch, On the combinatorial properties of bihyperbolic balancing number, Tatra Mt. Math. Publ. 77 (2020), 27–38.
  • [9] D. Bro ́d, A. Szynal-Liana, I. Włoch, On some combinatorial properties of bihyperbolic numbers of the Fibonacci type, Math. Methods Appl. Sci. Math. Methods Appl. Sci. 44(6) (2021), 4607–4615.
  • [10] J. Cockle, On certain functions resembling quaternions, and on a new imaginary in algebra, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 33 (1848), no. 224. 435–439.
  • [11] J. Cockle, On a new imaginary in algebra, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 34 (1849), no. 226. 37–47.
  • [12] J. Cockle, On the symbols of algebra and on the theory of Tessarines, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 34 (1849), no. 231. 406–410.
  • [13] M. Bilgin, S. Ersoy, Algebraic properties of bihyperbolic numbers, Adv. Appl. Clifford Alg. 30 (2020), no. 13.
  • [14] S. Ersoy, M. Bilgin, Topolojik Bihiperbolik Modu ̈ller (Turkish) [Topological Bihyperbolic Modules], 31. National Mathe- matics Symposium, Erzincan Binali Yıldırım University, Erzincan, Turkey, 2018, pp. 69.
  • [15] M.E. Luna Elizarrara ́s, M. Shapiro, C.O. Perez-Regalado, On linear functionals and Hahn-Banach theorems for hyperbolic and bicomplex modules, Adv. Appl. Clifford Alg. 24 (2014), 1105–1129.
  • [16] M.E. Luna Elizarrara ́s, M. Panza, M. Shapiro, D.C. Struppa, Geometry and Identity Theorems for Bicomplex Functions and Functions of a Hyperbolic Variable, Milan J. Math. 88 (2020), 247–261.
  • [17] R. Kumar, H. Saini, On Hahn Banach separation theorem for topological hyperbolic and topological bicomplex modules, arXiv preprint arXiv:1510.01538, 2015.
  • [18] R. Kumar, H. Saini, Topological bicomplex modules, Adv. Appl. Clifford Alg. 26 (2016), no. 4, 1249–1270.
  • [19] R. Larsen, Functional analysis, Marcel Dekker, New York, 1973.
  • [20] S. Olario, Complex numbers in n dimensions, North-Holland Mathematics Studies, Elsevier, vol. 190, 2002.
  • [21] W. Rudin, Functional analysis, 2nd Edition, McGraw Hill, New York, 1991.

Topological Bihyperbolic Modules

Year 2021, Volume: 4 Issue: 3, 115 - 129, 30.09.2021

Merve Bilgin Soley Ersoy

https://doi.org/10.33434/cams.985772

Abstract

The aim of this article is introducing and researching hyperbolic modules, bihyperbolic modules, topological hyperbolic modules, and topological bihyperbolic modules. In this regard, we define balanced, convex and absorbing sets in hyperbolic and bihyperbolic modules. In particular, we investigate convex sets in hyperbolic numbers set (it is a hyperbolic module over itself) by considering the isomorphic relation of this set with 2−2−dimensional Minkowski space. Moreover, bihyperbolic numbers set is a bihyperbolic module over itself, too. So, we define convex sets in this module by considering hypersurfaces of 4−4−dimensional semi Euclidean space that are isomorphic to some subsets of bihyperbolic numbers set. We also study the interior and closure of some special sets and neighbourhoods of the unit element of the module in the introduced topological bihyperbolic modules. In the light of obtained results, new relationships are presented for idempotent representations in topological bihyperbolic modules

Keywords

Bihyperbolic numbers hyperbolic numbers topological bihyperbolic modules topological modules

References

  • [1] A.A. Pogorui, R.M. Rodriguez-Dagnino, R.D. Rodrigue-Said, On the set of zeros of bihyperbolic polynomials, Complex Var. Elliptic Equ., 53 (2008), no. 7, 685–690.
  • [2] A. Grothendieck, Topological vector spaces, Gordon and Breach Science Publishers, New York, 1973.
  • [3] C. Segre, Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici (The real representation of complex elements and hyperalgebraic entities), Math. Annalen, 40 (1892), no. 3, 413–467.
  • [4] D. Alfsmann, H.G. Gockler, Hypercomplex bark-scale filter bank design based on allpass-phase specifications, Conference paper: Signal processing conference (EUSIPCO), Proceedings of the 20th European, Bucharest, Romania, 2012.
  • [5] D. Alpay, M.E. Luna Elizarraras, M. Shapiro, D.C. Struppa, Basics of functional analysis with bicomplex scalars and bicomplex Schur analysis, Springer Briefs in Mathematics, 2014.
  • [6] F. Catoni, D. Boccaletti, R. Cannata, ,V. Catoni, E. Nichelatti, P. Zampetti, The mathematics of Minkowski Space-Time with an introduction to commutative hypercomplex numbers, Birkhauser Verlag, Basel, Boston, Berlin, 2008.
  • [7] G. Baley Price, An introduction to multicomplex spaces and functions, Marcel Dekker Inc., New York, 1991.
  • [8] D. Bro ́d, A. Szynal-Liana, I. Włoch, On the combinatorial properties of bihyperbolic balancing number, Tatra Mt. Math. Publ. 77 (2020), 27–38.
  • [9] D. Bro ́d, A. Szynal-Liana, I. Włoch, On some combinatorial properties of bihyperbolic numbers of the Fibonacci type, Math. Methods Appl. Sci. Math. Methods Appl. Sci. 44(6) (2021), 4607–4615.
  • [10] J. Cockle, On certain functions resembling quaternions, and on a new imaginary in algebra, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 33 (1848), no. 224. 435–439.
  • [11] J. Cockle, On a new imaginary in algebra, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 34 (1849), no. 226. 37–47.
  • [12] J. Cockle, On the symbols of algebra and on the theory of Tessarines, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 34 (1849), no. 231. 406–410.
  • [13] M. Bilgin, S. Ersoy, Algebraic properties of bihyperbolic numbers, Adv. Appl. Clifford Alg. 30 (2020), no. 13.
  • [14] S. Ersoy, M. Bilgin, Topolojik Bihiperbolik Modu ̈ller (Turkish) [Topological Bihyperbolic Modules], 31. National Mathe- matics Symposium, Erzincan Binali Yıldırım University, Erzincan, Turkey, 2018, pp. 69.
  • [15] M.E. Luna Elizarrara ́s, M. Shapiro, C.O. Perez-Regalado, On linear functionals and Hahn-Banach theorems for hyperbolic and bicomplex modules, Adv. Appl. Clifford Alg. 24 (2014), 1105–1129.
  • [16] M.E. Luna Elizarrara ́s, M. Panza, M. Shapiro, D.C. Struppa, Geometry and Identity Theorems for Bicomplex Functions and Functions of a Hyperbolic Variable, Milan J. Math. 88 (2020), 247–261.
  • [17] R. Kumar, H. Saini, On Hahn Banach separation theorem for topological hyperbolic and topological bicomplex modules, arXiv preprint arXiv:1510.01538, 2015.
  • [18] R. Kumar, H. Saini, Topological bicomplex modules, Adv. Appl. Clifford Alg. 26 (2016), no. 4, 1249–1270.
  • [19] R. Larsen, Functional analysis, Marcel Dekker, New York, 1973.
  • [20] S. Olario, Complex numbers in n dimensions, North-Holland Mathematics Studies, Elsevier, vol. 190, 2002.
  • [21] W. Rudin, Functional analysis, 2nd Edition, McGraw Hill, New York, 1991.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Merve Bilgin This is me 0000-0003-2242-8940

Soley Ersoy 0000-0002-7183-7081

Publication Date September 30, 2021
Submission Date August 22, 2021
Acceptance Date October 1, 2021
Published in Issue Year 2021 Volume: 4 Issue: 3

Cite

APA Bilgin, M., & Ersoy, S. (2021). Topological Bihyperbolic Modules. Communications in Advanced Mathematical Sciences, 4(3), 115-129. https://doi.org/10.33434/cams.985772
AMA Bilgin M, Ersoy S. Topological Bihyperbolic Modules. Communications in Advanced Mathematical Sciences. September 2021;4(3):115-129. doi:10.33434/cams.985772
Chicago Bilgin, Merve, and Soley Ersoy. “Topological Bihyperbolic Modules”. Communications in Advanced Mathematical Sciences 4, no. 3 (September 2021): 115-29. https://doi.org/10.33434/cams.985772.
EndNote Bilgin M, Ersoy S (September 1, 2021) Topological Bihyperbolic Modules. Communications in Advanced Mathematical Sciences 4 3 115–129.
IEEE M. Bilgin and S. Ersoy, “Topological Bihyperbolic Modules”, Communications in Advanced Mathematical Sciences, vol. 4, no. 3, pp. 115–129, 2021, doi: 10.33434/cams.985772.
ISNAD Bilgin, Merve - Ersoy, Soley. “Topological Bihyperbolic Modules”. Communications in Advanced Mathematical Sciences 4/3 (September 2021), 115-129. https://doi.org/10.33434/cams.985772.
JAMA Bilgin M, Ersoy S. Topological Bihyperbolic Modules. Communications in Advanced Mathematical Sciences. 2021;4:115–129.
MLA Bilgin, Merve and Soley Ersoy. “Topological Bihyperbolic Modules”. Communications in Advanced Mathematical Sciences, vol. 4, no. 3, 2021, pp. 115-29, doi:10.33434/cams.985772.
Vancouver Bilgin M, Ersoy S. Topological Bihyperbolic Modules. Communications in Advanced Mathematical Sciences. 2021;4(3):115-29.
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