$mathcal{Z^ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure

Yıl 2024, Cilt: 7 Sayı: 2, 83 – 92, 23.05.2024

Abhishek Singh , S. K. Chaubey , Sunil Yadav , Shraddha Patel

https://doi.org/10.32323/ujma.1418496

Öz

The main goal of this manuscript is to investigate the properties of $N(k)$-contact metric manifolds admitting a $mathcal{Z^ast}$-tensor. We prove the necessary conditions for which $N(k)$-contact metric manifolds endowed with a $mathcal{Z^ast}$-tensor are Einstein manifolds. In this sequel, we accomplish that an $N(k)$-contact metric manifold endowed with a $mathcal{Z^ast}$-tensor satisfying $mathcal{Z^ast}(mathcal{G}_{1},hat{zeta})cdot mathcal{overset{star}R}=0$ is either locally isometric to the Riemannian product $E^{n+1}(0)times S^{n}(4)$ or an Einstein manifold. We also prove the condition for which an $N(k)$-contact metric manifold endowed with a $mathcal{Z^ast}$-tensor is a Sasakian manifold. To validate some of our results, we construct a non-trivial example of an $N(k)$-contact metric manifold.

Anahtar Kelimeler

$N(k)$-contact metric manifold, Einstein manifold, Ricci soliton, $mathcal{Z}^star$-recurrent, $mathcal{Z^ast}$-tensor

Kaynakça

  • [1] S. Tanno, Ricci curvatures of contact Riemannian manifolds, Tohoku Math. J., 40 (1988), 441–448.
  • [2] D. E. Blair, T. Koufogiorgos, B. J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Isr. J. Math., 91 (1995), 189–214.
  • [3] U. C. De, Y. J. Suh, S. K. Chaubey, Conformal vector fields on almost co-K¨ahler manifolds, Math. Slovaca, 71(6) (2021), 1545–1552.
  • [4] U. C. De, S. K. Chaubey, Y. J. Suh, A note on almost co-K¨ahler manifolds, Int. J. Geom. Methods Mod. Phys., 17(10) (2020), 2050153, 14 pp.
  • [5] S. K. Chaubey, M. A. Khan, A. S. R. Al Kaabi, N(k)-paracontact metric manifolds admitting the Fischer-Marsden conjecture, AIMS Math., 9(1) (2024), 2232–2243.
  • [6] S. K. Chaubey, K. K. Bhaishya, M. D. Siddiqi, Existence of some classes of N(k)-quasi Einstein manifolds, Bol. Soc. Parana. Mat., 39(5) (2021), 145–162.
  • [7] S. K. Chaubey, Certain results on N(k)-quasi Einstein manifolds, Afr. Mat., 30(1-2) (2019), 113–127.
  • [8] S. K. Yadav, S. K. Chaubey, D. L. Suthar, Certain results on almost Kenmotsu (k;m;n)􀀀spaces, Konuralp J. Math., 6(1) (2018) 128–133.
  • [9] H. İ. Yoldas, E. Yasar, A study on N(k)-contact metric manifolds, Balk. J. Geom. Its Appl., 25(1) (2020), 127–140.
  • [10] S. K. Yadav, X. Chen, On h-Einstein N(k)-contact metric manifolds, Bol. Soc. Pran. Mat., 41(3) (2021), 1–13.
  • [11] H. I. Yoldas, Certain Results on N(k)-Contact Metric Manifolds and Torse-Forming Vector Fields, J. Math. Ext., 15 (2021), 1–16.
  • [12] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differ. Geom., 17 (1982), 255—306.
  • [13] R. S. Hamilton, The Ricci flow on surfaces, Mathematics and General Relativity, Contemp. Math., Santa Cruz, CA, 1986, 71, Amer. Math. Soc. Providence, RI, (1988), 237–262.
  • [14] T. Chave, G. Valent, Quasi-Einstein metrics and their renormalizability properties, Helv. Phys. Acta, 69 (1996), 344–347.
  • [15] T. Chave, G. Valent, On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties, Nuclear Phys., 478 (1996), 758–778.
  • [16] G. Ayar, M. Yildirim, h-Ricci solitons on nearly Kenmotsu manifolds, Asian-Eur. J. Math., 12(6) (2019), 2040002.
  • [17] C. Calin, M. Crasmareanu, From the Eisenhart problem to Ricci solitons in f -Kenmotsu manifolds, Bull. Malyas. Math. Sci. Soc., 33 (2010), 361–368.
  • [18] Y. J. Suh, S. K. Chaubey, MNI Khan, Lorentzian manifolds: A characterization with a type of semi-symmetric non-metric connection, Rev. Math. Phys., 36(3) (2024), Paper No. 2450001.
  • [19] S. K. Chaubey, U. C. De, Y. J. Suh, Conformal vector field and gradient Einstein solitons on h-Einstein cosymplectic manifolds, Int. J. Geom. Methods Mod. Phys., 20(8) (2023), Paper No. 2350135, 16 pp.
  • [20] Y. J. Suh, S. K. Chaubey, Ricci solitons on general relativistic spacetimes, Phys. Scr., 98 (2023), 065207.
  • [21] A. Haseeb, S. K. Chaubey, M. A. Khan, Riemannian 3-manifolds and Ricci-Yamabe solitons, Int. J. Geom. Methods Mod. Phys., 20(1) (2023), Paper No. 2350015, 13 pp.
  • [22] S. K. Chaubey, Y. J. Suh, Riemannian concircular structure manifolds, Filomat, 36(19) (2022), 6699–6711.
  • [23] S. K. Chaubey, G. -E. Vˆılcu, Gradient Ricci solitons and Fischer-Marsden equation on cosymplectic manifolds, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 116(4) (2022), Paper No. 186, 14 pp.
  • [24] A. Haseeb, R. Prasad, Some results on Lorentzian para-Kenmotsu manifolds, Bulletin of the Transilvania University of Brasov, Series III: Mathematics, Informatics, Physics, 13(62) (2020), 185–198.
  • [25] A. Haseeb, S. K. Chaubey, Lorentzian para-Sasakian manifolds and ?-Ricci solitons, Kragujev. J. Math., 48(2) (2024), 167–179.
  • [26] H. Öztürk, S. K. Yadav, A note on Ricci and Yamabe solitons on almost Kenmotsu manifolds, Novi Sad J. Math., 53(2) (2023), 223–239.
  • [27] A. Sarkar, G. G. Biswas, Ricci solitons on three-dimensional generalized Sasakian-space forms with quasi-Sasakian metric, Africa Maths., 31 (2020), 455–463.
  • [28] H. İ. Yoldaş, A. Haseeb, F. Mofarreh, Certain curvature conditions on Kenmotsu manifolds and ?􀀀h-Ricci solitons, Axioms, 12(2) (2023), 14 pages.
  • [29] C. A. Mantica, L. G. Molonari, Weakly Z -symmetric manifolds, Acta Math. Hunger., 135 (2012), 80–96.
  • [30] M. Ali, A. Haseeb, F. Mofarreh, M. Vasiulla, Z -symmetric manifolds admitting Schouten tensor, Mathematics, 10 (2022), 4293, https://doi.org/10.3390/math10224293.
  • [31] S. K. Chaubey, Trans-Sasakian manifolds satisfying certain conditions, TWMS J. App. Eng. Math., 9(2) (2019), 305–314.
  • [32] S. K. Chaubey, On special weakly Riccisymmetric and generalized Ricci-recurrent trans-Sasakian structures, Thai J. Math., 16(3) (2018), 693–707.
  • [33] A. Barman, I. Unal, Geometry of Kenmotsu manifolds admitting Z-tensor, Bull. Transilv. Univ. Bras., 2(64) (2022), 23–40.
  • [34] U. S. Negi, P. Chauhan, Tensor structures and recurrent Z -forms in Riemannian manifolds, Aryabhatta J. Math. Inf., 14(2) (2022), 153–160.
  • [35] D. G. Prakasha, P. Veeresha, M. Nagaraja, Z -symmetries of e-para-Sasakian 3-manifolds, arXiv:1909.05535v1, (2019).
  • [36] I. Unal, N(k)-contact metric manifolds admitting Z -tensor, KMU J. Eng. Natural Sciences, 2(1) (2020), 64–69.
  • [37] A. L. Besse, Einstein manifolds, Ergeb. Math. Grenzgeb., 3. Folge, Bd. 10, Springer-Verlag, Berlin, Heidelberg, New York, (1987).
  • [38] A. Derdzinski, C. L. Shen, Codazzi tensor fields curvature and Pontryagin forms, Proc. Lond. Math. Soc., 47(1) (1983), 15-–26.
  • [39] F. de Felice, C. J. S. Clarke, Relativity on Curved Manifolds, Cambridge University Press, Cambridge, (1990).
  • [40] C. A. Mantica, Y. J. Suh, Pseudo-Q-symmetric Riemannian manifolds, Int. J. Geom. Methods Mod. Phys., 10(5) (2013), 25 pages.
  • [41] K. Yano, Concircular geometry I, Concircular transformation, Proc. Imp. Acad. Tokyo, 16 (1940), 195-200.
  • [42] S. K. Yadav, A. Yildiz, Q-curvature tensor on f -Kenmotsu 3-manifolds, Univers. J. Math. Appl., 5(3) (2022), 96-106.
  • [43] M. Yildirim, A new characterization of Kenmotsu manifolds with respect to Q tensor, J. Geom. Phys., 176 (2022), 104498.
  • [44] D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes In Math., 509, Springer-Verlag Berlin Heidelberg, (1976).
  • [45] S. K. Yadav, X. Chen, A note on (k;m)-contact metric manifolds, Analele University Oradea Fasc. Matematica, XXIX(2) (2022), 17–28.
  • [46] B. J. Papantoniou, Contact Riemannian manifolds satisfying R(x ;X):R=0 and x 2 (k;m)-nullity distribution, Yokohama Math. J., 40 (1993), 149–161.
  • [47] D. E. Blair, J. S. Kim, M. M. Tripathi, On the concircular curvature tensor of a contact metric manifold, Fundam. J. Math. Appl., 3(2) (2020), 94–100.
  • [48] D. E. Blair, Two remarks on contact metric structures, Tohoku Math. J., 40 (1988), 441-448.
  • [49] C. A. Mantica, Y. J. Suh, Pseudo Z- symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys., 9(1) (2012), 1250004.
  • [50] C. A. Mantica, Y. J. Suh, Pseudo Z-symmetric space-times, J. Math. Phys., 55 (2014), 042502.
  • [51] C. A. Mantica, Y. J. Suh, Recurrent Z-forms on Riemannian and Kaehler manifolds, Int. J. Geom. Methods Mod. Phys., 9 (2012), 1250059.

Yıl 2024, Cilt: 7 Sayı: 2, 83 – 92, 23.05.2024

Abhishek Singh , S. K. Chaubey , Sunil Yadav , Shraddha Patel

https://doi.org/10.32323/ujma.1418496

Öz

Kaynakça

  • [1] S. Tanno, Ricci curvatures of contact Riemannian manifolds, Tohoku Math. J., 40 (1988), 441–448.
  • [2] D. E. Blair, T. Koufogiorgos, B. J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Isr. J. Math., 91 (1995), 189–214.
  • [3] U. C. De, Y. J. Suh, S. K. Chaubey, Conformal vector fields on almost co-K¨ahler manifolds, Math. Slovaca, 71(6) (2021), 1545–1552.
  • [4] U. C. De, S. K. Chaubey, Y. J. Suh, A note on almost co-K¨ahler manifolds, Int. J. Geom. Methods Mod. Phys., 17(10) (2020), 2050153, 14 pp.
  • [5] S. K. Chaubey, M. A. Khan, A. S. R. Al Kaabi, N(k)-paracontact metric manifolds admitting the Fischer-Marsden conjecture, AIMS Math., 9(1) (2024), 2232–2243.
  • [6] S. K. Chaubey, K. K. Bhaishya, M. D. Siddiqi, Existence of some classes of N(k)-quasi Einstein manifolds, Bol. Soc. Parana. Mat., 39(5) (2021), 145–162.
  • [7] S. K. Chaubey, Certain results on N(k)-quasi Einstein manifolds, Afr. Mat., 30(1-2) (2019), 113–127.
  • [8] S. K. Yadav, S. K. Chaubey, D. L. Suthar, Certain results on almost Kenmotsu (k;m;n)􀀀spaces, Konuralp J. Math., 6(1) (2018) 128–133.
  • [9] H. İ. Yoldas, E. Yasar, A study on N(k)-contact metric manifolds, Balk. J. Geom. Its Appl., 25(1) (2020), 127–140.
  • [10] S. K. Yadav, X. Chen, On h-Einstein N(k)-contact metric manifolds, Bol. Soc. Pran. Mat., 41(3) (2021), 1–13.
  • [11] H. I. Yoldas, Certain Results on N(k)-Contact Metric Manifolds and Torse-Forming Vector Fields, J. Math. Ext., 15 (2021), 1–16.
  • [12] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differ. Geom., 17 (1982), 255—306.
  • [13] R. S. Hamilton, The Ricci flow on surfaces, Mathematics and General Relativity, Contemp. Math., Santa Cruz, CA, 1986, 71, Amer. Math. Soc. Providence, RI, (1988), 237–262.
  • [14] T. Chave, G. Valent, Quasi-Einstein metrics and their renormalizability properties, Helv. Phys. Acta, 69 (1996), 344–347.
  • [15] T. Chave, G. Valent, On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties, Nuclear Phys., 478 (1996), 758–778.
  • [16] G. Ayar, M. Yildirim, h-Ricci solitons on nearly Kenmotsu manifolds, Asian-Eur. J. Math., 12(6) (2019), 2040002.
  • [17] C. Calin, M. Crasmareanu, From the Eisenhart problem to Ricci solitons in f -Kenmotsu manifolds, Bull. Malyas. Math. Sci. Soc., 33 (2010), 361–368.
  • [18] Y. J. Suh, S. K. Chaubey, MNI Khan, Lorentzian manifolds: A characterization with a type of semi-symmetric non-metric connection, Rev. Math. Phys., 36(3) (2024), Paper No. 2450001.
  • [19] S. K. Chaubey, U. C. De, Y. J. Suh, Conformal vector field and gradient Einstein solitons on h-Einstein cosymplectic manifolds, Int. J. Geom. Methods Mod. Phys., 20(8) (2023), Paper No. 2350135, 16 pp.
  • [20] Y. J. Suh, S. K. Chaubey, Ricci solitons on general relativistic spacetimes, Phys. Scr., 98 (2023), 065207.
  • [21] A. Haseeb, S. K. Chaubey, M. A. Khan, Riemannian 3-manifolds and Ricci-Yamabe solitons, Int. J. Geom. Methods Mod. Phys., 20(1) (2023), Paper No. 2350015, 13 pp.
  • [22] S. K. Chaubey, Y. J. Suh, Riemannian concircular structure manifolds, Filomat, 36(19) (2022), 6699–6711.
  • [23] S. K. Chaubey, G. -E. Vˆılcu, Gradient Ricci solitons and Fischer-Marsden equation on cosymplectic manifolds, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 116(4) (2022), Paper No. 186, 14 pp.
  • [24] A. Haseeb, R. Prasad, Some results on Lorentzian para-Kenmotsu manifolds, Bulletin of the Transilvania University of Brasov, Series III: Mathematics, Informatics, Physics, 13(62) (2020), 185–198.
  • [25] A. Haseeb, S. K. Chaubey, Lorentzian para-Sasakian manifolds and ?-Ricci solitons, Kragujev. J. Math., 48(2) (2024), 167–179.
  • [26] H. Öztürk, S. K. Yadav, A note on Ricci and Yamabe solitons on almost Kenmotsu manifolds, Novi Sad J. Math., 53(2) (2023), 223–239.
  • [27] A. Sarkar, G. G. Biswas, Ricci solitons on three-dimensional generalized Sasakian-space forms with quasi-Sasakian metric, Africa Maths., 31 (2020), 455–463.
  • [28] H. İ. Yoldaş, A. Haseeb, F. Mofarreh, Certain curvature conditions on Kenmotsu manifolds and ?􀀀h-Ricci solitons, Axioms, 12(2) (2023), 14 pages.
  • [29] C. A. Mantica, L. G. Molonari, Weakly Z -symmetric manifolds, Acta Math. Hunger., 135 (2012), 80–96.
  • [30] M. Ali, A. Haseeb, F. Mofarreh, M. Vasiulla, Z -symmetric manifolds admitting Schouten tensor, Mathematics, 10 (2022), 4293, https://doi.org/10.3390/math10224293.
  • [31] S. K. Chaubey, Trans-Sasakian manifolds satisfying certain conditions, TWMS J. App. Eng. Math., 9(2) (2019), 305–314.
  • [32] S. K. Chaubey, On special weakly Riccisymmetric and generalized Ricci-recurrent trans-Sasakian structures, Thai J. Math., 16(3) (2018), 693–707.
  • [33] A. Barman, I. Unal, Geometry of Kenmotsu manifolds admitting Z-tensor, Bull. Transilv. Univ. Bras., 2(64) (2022), 23–40.
  • [34] U. S. Negi, P. Chauhan, Tensor structures and recurrent Z -forms in Riemannian manifolds, Aryabhatta J. Math. Inf., 14(2) (2022), 153–160.
  • [35] D. G. Prakasha, P. Veeresha, M. Nagaraja, Z -symmetries of e-para-Sasakian 3-manifolds, arXiv:1909.05535v1, (2019).
  • [36] I. Unal, N(k)-contact metric manifolds admitting Z -tensor, KMU J. Eng. Natural Sciences, 2(1) (2020), 64–69.
  • [37] A. L. Besse, Einstein manifolds, Ergeb. Math. Grenzgeb., 3. Folge, Bd. 10, Springer-Verlag, Berlin, Heidelberg, New York, (1987).
  • [38] A. Derdzinski, C. L. Shen, Codazzi tensor fields curvature and Pontryagin forms, Proc. Lond. Math. Soc., 47(1) (1983), 15-–26.
  • [39] F. de Felice, C. J. S. Clarke, Relativity on Curved Manifolds, Cambridge University Press, Cambridge, (1990).
  • [40] C. A. Mantica, Y. J. Suh, Pseudo-Q-symmetric Riemannian manifolds, Int. J. Geom. Methods Mod. Phys., 10(5) (2013), 25 pages.
  • [41] K. Yano, Concircular geometry I, Concircular transformation, Proc. Imp. Acad. Tokyo, 16 (1940), 195-200.
  • [42] S. K. Yadav, A. Yildiz, Q-curvature tensor on f -Kenmotsu 3-manifolds, Univers. J. Math. Appl., 5(3) (2022), 96-106.
  • [43] M. Yildirim, A new characterization of Kenmotsu manifolds with respect to Q tensor, J. Geom. Phys., 176 (2022), 104498.
  • [44] D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes In Math., 509, Springer-Verlag Berlin Heidelberg, (1976).
  • [45] S. K. Yadav, X. Chen, A note on (k;m)-contact metric manifolds, Analele University Oradea Fasc. Matematica, XXIX(2) (2022), 17–28.
  • [46] B. J. Papantoniou, Contact Riemannian manifolds satisfying R(x ;X):R=0 and x 2 (k;m)-nullity distribution, Yokohama Math. J., 40 (1993), 149–161.
  • [47] D. E. Blair, J. S. Kim, M. M. Tripathi, On the concircular curvature tensor of a contact metric manifold, Fundam. J. Math. Appl., 3(2) (2020), 94–100.
  • [48] D. E. Blair, Two remarks on contact metric structures, Tohoku Math. J., 40 (1988), 441-448.
  • [49] C. A. Mantica, Y. J. Suh, Pseudo Z- symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys., 9(1) (2012), 1250004.
  • [50] C. A. Mantica, Y. J. Suh, Pseudo Z-symmetric space-times, J. Math. Phys., 55 (2014), 042502.
  • [51] C. A. Mantica, Y. J. Suh, Recurrent Z-forms on Riemannian and Kaehler manifolds, Int. J. Geom. Methods Mod. Phys., 9 (2012), 1250059.

Toplam 51 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Topoloji
BölümMakaleler
Yazarlar

Abhishek Singh Dr. Rammanohar Lohia Avadh University, Ayodhya-224001, Uttar Pradesh, India. 0009-0007-6784-7395 India

S. K. Chaubey Banaras Hindu University, University of Technology and Applied Sciences-Shinas 0000-0002-3882-4596 Oman

Sunil Yadav United College of Engineering & Research, A-31, UPSIDC Institutional Area, Naini-211010, Prayagraj, Uttar Pradesh, India. 0000-0001-6930-3585 India

Shraddha Patel Dr. Rammanohar Lohia Avadh University, Ayodhya (UP) 0000-0001-9773-9546 India

Erken Görünüm Tarihi11 Mayıs 2024
Yayımlanma Tarihi23 Mayıs 2024
Gönderilme Tarihi12 Ocak 2024
Kabul Tarihi11 Nisan 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 7 Sayı: 2

Kaynak Göster

APASingh, A., Chaubey, S. K., Yadav, S., Patel, S. (2024). $mathcal{Z^ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure. Universal Journal of Mathematics and Applications, 7(2), 83-92. https://doi.org/10.32323/ujma.1418496
AMASingh A, Chaubey SK, Yadav S, Patel S. $mathcal{Z^ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure. Univ. J. Math. Appl. Mayıs 2024;7(2):83-92. doi:10.32323/ujma.1418496
ChicagoSingh, Abhishek, S. K. Chaubey, Sunil Yadav, ve Shraddha Patel. “$mathcal{Z^ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure”. Universal Journal of Mathematics and Applications 7, sy. 2 (Mayıs 2024): 83-92. https://doi.org/10.32323/ujma.1418496.
EndNoteSingh A, Chaubey SK, Yadav S, Patel S (01 Mayıs 2024) $mathcal{Z^ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure. Universal Journal of Mathematics and Applications 7 2 83–92.
IEEEA. Singh, S. K. Chaubey, S. Yadav, ve S. Patel, “$mathcal{Z^ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure”, Univ. J. Math. Appl., c. 7, sy. 2, ss. 83–92, 2024, doi: 10.32323/ujma.1418496.
ISNADSingh, Abhishek vd. “$mathcal{Z^ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure”. Universal Journal of Mathematics and Applications 7/2 (Mayıs 2024), 83-92. https://doi.org/10.32323/ujma.1418496.
JAMASingh A, Chaubey SK, Yadav S, Patel S. $mathcal{Z^ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure. Univ. J. Math. Appl. 2024;7:83–92.
MLASingh, Abhishek vd. “$mathcal{Z^ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure”. Universal Journal of Mathematics and Applications, c. 7, sy. 2, 2024, ss. 83-92, doi:10.32323/ujma.1418496.
VancouverSingh A, Chaubey SK, Yadav S, Patel S. $mathcal{Z^ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure. Univ. J. Math. Appl. 2024;7(2):83-92.

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