User:Jacobolus/Double


 * Resources about the "double numbers", a.k.a. "hyperbolic numbers", "split-complex numbers", "perplex numbers", ...


 * Reany, Patrick. "Complex clifford algebra and Nth-order linear differential equations." Advances in Applied Clifford Algebras 3 (1993): 121-127.

real tessarines:
 * Cockle, James (1849) On a New Imaginary in Algebra 34:37–47, London-Edinburgh-Dublin Philosophical Magazine (3) 33:435–9

pseudo-complex numbers:
 * Schäfer, M., Hess, P. O., & Greiner, W. (2014). Geometry of pseudo-complex General Relativity. Astronomische Nachrichten, 335(6-7), 751–756. doi:10.1002/asna.201412104
 * Hess, Peter O., Leila Maghlaoui, and Walter Greiner. "The Robertson–Walker metric in a pseudo-complex general relativity." International Journal of Modern Physics E 19, no. 07 (2010): 1315-1339. doi:10.1142/S021830131001576X

semi-complex numbers:
 * Antonuccio, Francesco (1993). "Semi-Complex Analysis & Mathematical Physics" arXiv:gr-qc/9311032

hypercomplex numbers:
 * Azri, Hemza, and A. Bounames. "Geometrical origin of the cosmological constant." General Relativity and Gravitation 44 (2012): 2547-2561. 10.1007/s10714-012-1413-9

bireal numbers:
 * Bencivenga, Uldrico (1946) "Sulla rappresentazione geometrica delle algebre doppie dotate di modulo", Atti della Reale Accademia delle Scienze e Belle-Lettere di Napoli, Ser (3) v.2 No7.

anormal-complex numbers:
 * W. Benz (1973). Vorlesungen über Geometrie der Algebren, Springer, Berlin.

spacetime numbers:
 * N. Borota, E. Flores and T. J. Osler (2000). Spacetime numbers the easy way, Mathematics and Computer Education, Vol. 34, No. 2, pp. 159-168.

duplex numbers:
 * Kocik, Jerzy (1999). "Duplex numbers, diffusion systems, and generalized quantum mechanics." International Journal of Theoretical Physics 38, no. 8: 2221-2230. doi:10.1023/A:1026678408244 (arXiv version has: Although the text is not altered, the author now prefers term “hyperbolic numbers” over the original ”duplex numbers”)
 * Kocik, Jerzy (2013). "A porism concerning cyclic quadrilaterals." Geometry 2013. doi:10.1155/2013/483727

paracomplex numbers:
 * Cruceanu, Vasile, Pedro Fortuny, and Pedro M. Gadea. "A survey on paracomplex geometry." The Rocky Mountain Journal of Mathematics 26, no. 1 (1996): 83-115. https://core.ac.uk/download/pdf/36026973.pdf
 * Cartas-Fuentevilla, R., and O. Meza-Aldama (2016). "Spontaneous symmetry breaking, and strings defects in hypercomplex gauge field theories." The European Physical Journal C 76: 1-22. doi:10.1140/epjc/s10052-016-3944-9
 * Médevielle, M., T. Mohaupt, and G. Pope (2022). "Type-II Calabi-Yau compactifications, T-duality and special geometry in general spacetime signature." Journal of High Energy Physics 2022, no. 2: 1-42. doi:10.1007/JHEP02(2022)048

countercomplex numbers:
 * Musès, C. (1977). Applied hypernumbers: computational concepts. Applied Mathematics and Computation, 3(3), 211–226. doi:10.1016/0096-3003(77)90002-9
 * Musès, C. (1980). Hypernumbers and quantum field theory with a summary of physically applicable hypernumber arithmetics and their geometries. Applied Mathematics and Computation, 6(1), 63–94. doi:10.1016/0096-3003(80)90016-8
 * Carmody, K. (1988). Circular and hyperbolic quaternions, octonions, and sedenions. Applied Mathematics and Computation, 28(1), 47-72.
 * Carmody, K. (1997). Circular and hyperbolic quaternions, octonions, and sedenions—Further results. Applied Mathematics and Computation, 84(1), 27–47. doi:10.1016/s0096-3003(96)00051-3

perplex numbers:
 * P. Fjelstad, Extending special relativity via the perplex numbers, Am. J. Phys. 54 (1986), pp. 416–422. doi:10.1119/1.14605
 * Band, William (1988). "Comments on ‘‘Extending relativity via the perplex numbers’’[Am. J. Phys. 5 4, 416 (1986)]." American Journal of Physics 56, no. 5: 469-469. doi:10.1119/1.15582
 * Assis, A. K. T. (1991). "Perplex numbers and quaternions." International Journal of Mathematical Education in Science and Technology 22, no. 4: 555-562. doi:10.1080/0020739910220406
 * Poodiack, Robert D., and Kevin J. LeClair (2009). "Fundamental theorems of algebra for the perplexes." The College Mathematics Journal 40, no. 5: 322-336. doi:10.4169/074683409X475643
 * Sporn, Howard (2017). "Pythagorean triples, complex numbers, and perplex numbers." The College Mathematics Journal 48, no. 2: 115-122.
 * Amorim, Ronni Geraldo Gomes de, Wytler Cordeiro dos Santos, Lindomar Bonfim Carvalho, and Ian Rodrigues Massa (2018). "A physical approach of perplex numbers." Revista Brasileira de Ensino de Física 40. doi:10.1590/1806-9126-RBEF-2017-0356
 * Ponomarenko, Vadim (2022). "Factorization of Perplex Integers." The American Mathematical Monthly 129, no. 6: 576-581. doi:10.1080/00029890.2022.2051406

Lorentz numbers:
 * Harvey, F. Reese (1990). Spinors and calibrations. Vol. 8. Elsevier.
 * Catoni, F. (1994). Riemann flat spaces, hypercomplex numbers and general relativity. No. ENEA-RT-ERG-94-05. P00024673. https://cds.cern.ch/record/266546/files/P00024673.pdf
 * Kimura, Makoto (2003). "Space of geodesics in hyperbolic spaces and Lorentz numbers." Mem. Fac. Sci. Eng. Shimane Univ. Ser. B Math. Sci 36: 61-67. http://www.math.shimane-u.ac.jp/memoir/36/bull1.pdf
 * Terlizzi, Luigia Di, Jerzy Julian Konderak, and Ignazio Lacirasella (2014). "On differentable functions over Lorentz numbers and their geometric applications." Differential Geometry--Dynamical Systems 16. http://www.mathem.pub.ro/dgds/v16/D16-di-890.pdf
 * Bayard, Pierre, and Victor Patty (2015). "Spinor representation of Lorentzian surfaces in R2, 2." Journal of Geometry and Physics 95: 74-95. doi:10.1016/j.geomphys.2015.05.002
 * Salvai, Marcos (2015). "Generalized geometric structures on complex and symplectic manifolds." Annali di Matematica Pura ed Applicata 194, no. 5: 1505-1525.
 * Da Silva, Luiz CB (2017). "Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spaces." arXiv preprint arXiv:1707.06321
 * Kassabov, Ognian, and Velichka Milousheva (2020). "Weierstrass Representations of Lorentzian Minimal Surfaces in R 2 4." Mediterranean Journal of Mathematics 17, no. 6: 199. doi:10.1007/s00009-020-01636-x
 * da Silva, Luiz CB (2021). "Surfaces of revolution with prescribed mean and skew curvatures in Lorentz-Minkowski space". Tohoku Math. J. 73 (2021) 317-319; doi:10.2748/tmj.20190729 arXiv:1804.00259

split-complex numbers
 * Inoguchi, Jun-ichi, and Magdalena Toda (2004). "Timelike minimal surfaces via loop groups." Acta Applicandae Mathematicae 83: 313-355. doi:10.1023/B:ACAP.0000039015.45368.f6 arxiv:0409073
 * Mondoc, Daniel (2007). "Compact exceptional simple Kantor triple systems defined on tensor products of composition algebras." Communications in Algebra 35, no. 11: 3699-3712. doi:10.1080/00927870701404739
 * Kunegis, Jérôme, Gerd Gröner, and Thomas Gottron (2012). "Online dating recommender systems: The split-complex number approach." In Proceedings of the 4th ACM RecSys workshop on Recommender systems and the social web, pp. 37-44. doi:10.1145/2365934.2365942
 * Petoukhov, S. V. (2012). "Symmetries of the genetic code, hypercomplex numbers and genetic matrices with internal complementarities." Symmetry: Culture and Science 23, no. 3-4: 275-301.
 * Gao, Changjun, Xuelei Chen, and You-Gen Shen (2016). "Quintessence and phantom emerging from the split-complex field and the split-quaternion field." General Relativity and Gravitation 48: 1-23.

split complex numbers
 * Rosenfeld, B. A. (1988). "Groups of Transformations." A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space: 327-381. doi:10.1007/978-1-4419-8680-1_9
 * Yokota, Ichiro. "Realizations of Involutive Automorphisms σ and Gσ of Exceptional Linear Lie Groups G, PART I, G= G₂. F₄ AND E₆." Tsukuba journal of mathematics 14, no. 1 (1990): 185-223.
 * Shaw, Ronald (1990). "Ternary composition algebras II. Automorphism groups and subgroups." Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 431, no. 1881: 21-36. doi:10.1098/rspa.1990.0116
 * Rosenfeld, Boris A. (1993). "Spaces with exceptional fundamental groups." Publications de l’Institut Mathématique, nouvelle série 54, no. 68: 97-119. http://elib.mi.sanu.ac.rs/files/journals/publ/74/n068p097.pdf
 * Rosenfeld, B. A. (1998). "Geometry of planes over nonassociative algebras." Acta Applicandae Mathematica 50: 103-110. doi:10.1023/A:1005871202247
 * Fischbacher, Thomas, Hermann Nicolai, and Henning Samtleben (2004). "Non-semisimple and complex gaugings of N= 16 supergravity." Communications in mathematical physics 249, no. 3: 475-496. doi:10.1007/s00220-004-1081-z
 * Matvei Libine (2007). Hyperbolic Cauchy Integral Formula for the Split Complex Numbers. arXiv:0712.0375.
 * Günaydin, Murat, and Oleksandr Pavlyk (2009). "Quasiconformal realizations of E6 (6), E7 (7), E8 (8) and SO (n+ 3, m+ 3), N≥ 4 supergravity and spherical vectors." Advances in Theoretical and Mathematical Physics 13, no. 6: 1895. arxiv:0904.0784
 * Boi, L. (2009). "Clifford geometric algebras, spin manifolds, and group actions in mathematics and physics". Advances in applied Clifford algebras, 19, pp. 611-656. doi:10.1007/s00006-009-0199-7
 * Sahay, Anurag, and Amitabh Virmani (2013). "Subtracted geometry from Harrison transformations: II." Journal of High Energy Physics 2013, no. 7: 1-16. doi:10.1007/JHEP07(2013)089
 * Emanuello, J. A., & Nolder, C. A. (2014). Projective Compactification of $$\R^{1,1}$$ and Its Möbius Geometry. Complex Analysis and Operator Theory, 9(2), 329–354. doi:10.1007/s11785-014-0363-5
 * Dray, Tevian, John Huerta, and Joshua Kincaid (2014). "The magic square of Lie groups: The 2×2 case." Letters in Mathematical Physics 104: 1445-1468. doi:10.1007/s11005-014-0720-3
 * Estévez, Pilar G., Francisco Jose Herranz, Javier de Lucas, and Cristina Sardón (2016). "Lie symmetries for Lie systems: applications to systems of ODEs and PDEs." Applied Mathematics and Computation 273: 435-452. doi:10.1016/j.amc.2015.09.078 arxiv:1404.2740
 * Wickstead, A. W. (2017). "Two dimensional unital Riesz algebras, their representations and norms." Positivity 21, no. 2: 787-801.
 * Gogioso, Stefano (2017). "Fantastic quantum theories and where to find them." arxiv:1703.10576
 * Fioresi, Rita, Emanuele Latini, and Alessio Marrani (2017). "Klein and conformal superspaces, split algebras and spinor orbits." Reviews in Mathematical Physics 29, no. 04: 1750011. doi:10.1142/S0129055X17500118
 * Ghiloni, Riccardo, Alessandro Perotti, and Caterina Stoppato (2017). "The algebra of slice functions." Transactions of the American Mathematical Society 369, no. 7: 4725-4762. doi:10.1090/tran/6816
 * Elliott, Jesse (2018). "Integer-valued polynomials on commutative rings and modules." Communications in Algebra 46, no. 3: 1121-1137. doi:10.1080/00927872.2017.1388811 arxiv:1608.00171
 * Couto, Ivo Terek, and Alexandre Lymberopoulos (2021). Introduction to Lorentz geometry: curves and surfaces. CRC Press.
 * Krasnov, Kirill (2022). "Spin (11, 3), particles, and octonions." Journal of Mathematical Physics 63, no. 3: 031701. doi:10.1063/5.0070058

complex hyperbolic numbers:
 * Akar, Mutlu, Salim Yüce, and S. Sahin (2018). "On the dual hyperbolic numbers and the complex hyperbolic numbers." Journal of Computer Science & Computational Mathematics 8, no. 1: 1-6. https://www.jcscm.net/fp/126.pdf

hyperbolic complex numbers:
 * Durañona Vedia A. and J. C. Vignaux (1935), “Theory of functions of a hyperbolic complex variable”, Publ, de Facultad Ciencias Fisiciomatematicas Contrib. (Universidad Nacional de La Plata - Argentina) 104: 139–183.
 * Vignaux, J.(1935) "Sobre el numero complejo hiperbolico y su relacion con la geometria de Borel", Contribucion al Estudio de las Ciencias Fisicas y Matematicas, Universidad Nacional de la Plata, Republica Argentina
 * Cree, George C. (1949). The Number Theory of a System of Hyperbolic Complex Numbers (MA thesis). McGill University.
 * Kunstatter, G., Moffat, J. W., & Malzan, J. (1983). Geometrical interpretation of a generalized theory of gravitation. Journal of Mathematical Physics, 24(4), 886–889. doi:10.1063/1.525777
 * Zhong, Z. (1985). Generation of new solutions of the stationary axisymmetric Einstein equations by a double complex function method. Journal of Mathematical Physics, 26(10), 2589–2595. doi:10.1063/1.526972
 * Lambert, D., & Tombal, P. (1987). Hyperbolic complex numbers and nonlinear sigma models. International Journal of Theoretical Physics, 26(10), 943–950. doi:10.1007/bf00670818
 * Piette, Bernard, and Wojciech J. Zakrzewski (1990). "Finite energy solutons for (1+ 1)‐dimensional σ models." Journal of Mathematical Physics 31, no. 4: 916-923. doi:10.1063/1.528772
 * Hucks, Joseph (1993). "Hyperbolic complex structures in physics." Journal of mathematical physics 34, no. 12: 5986-6008. doi:10.1063/1.530244
 * Gal, S. G. (1997). "Approximation and interpolation of functions of hyperbolic complex variable." Rev. Un. Mat. Argentina 40, no. 3-4: 25-35. https://www.inmabb.criba.edu.ar/revuma/pdf/v40n3y4/v40n3y4a03.pdf
 * Fjelstad, Paul, and Sorin G. Gal (2001). "Two-dimensional geometries, topologies, trigonometries and physics generated by complex-type numbers." Advances in Applied Clifford Algebras 11: 81-107. doi:10.1007/BF03042040
 * Apostolova, L. N., Krastev, K. I., Kiradjiev, B., Venkov, G., Kovacheva, R., & Pasheva, V. (2009). Hyperbolic double-complex numbers. AIP Conference Proceedings 1184, 193. doi:10.1063/1.3271614
 * Jakubska-Busse, Anna, M. W. Janowicz, Luiza Ochnio, and J. M. A. Ashbourn (2018). "Pickover biomorphs and non-standard complex numbers." Chaos, Solitons & Fractals 113: 46-52. doi:10.1016/j.chaos.2018.05.001
 * Veldsman, Stefan (2019). "Generalized complex numbers over near-fields." Quaestiones Mathematicae 42, no. 2: 181-200. doi:10.2989/16073606.2018.1442884

real hyperbolic numbers:
 * Smith, Norman E. (1949). Introduction to Hyperbolic Number Theory (MA thesis). McGill University.

hyperbolic numbers:
 * Miller, William, and Rochelle Boehning (1968). "Gaussian, parabolic, and hyperbolic numbers." The Mathematics Teacher 61, no. 4: 377-382. doi:10.5951/MT.61.4.0377
 * Lavrentiev, Mikhail, and Boris Chabat (1980). Effets hydrodynamiques et modèles mathématiques. Mir.
 * G. Sobczyk (1995). The hyperbolic number plane. The College Math. J., 26:268–280.
 * Smith, Tim (1996). "The Hyperbolic Number Plane and its Application to Lorentzian Geometry." https://inspire.redlands.edu/work/ns/e4eff62a-470b-4428-aca0-45338a0ea7b9
 * Motter, A. E., & Rosa, M. A. F. (1998). Hyperbolic Calculus. Advances in Applied Clifford Algebras, 8(1), 109–128. doi:10.1007/bf03041929
 * Antonuccio, Francesco (1998). "Hyperbolic numbers and the Dirac spinor." arXiv:hep-th/9812036
 * Buchholz, S., & Sommer, G. (2000). A hyperbolic multilayer perceptron. Proceedings of the IEEE-INNS-ENNS International Joint Conference on Neural Networks. IJCNN 2000. Neural Computing: New Challenges and Perspectives for the New Millennium. doi:10.1109/ijcnn.2000.857886
 * Gogberashvili, Merab (2002). "Observable algebra." arXiv:hep-th/0212251
 * Rochon, Dominic, and Michael Shapiro (2004). "On algebraic properties of bicomplex and hyperbolic numbers." Anal. Univ. Oradea, fasc. math 11, no. 71: 110. http://3dfractals.com/docs/Article01_bicomplex.pdf
 * Catoni, Francesco, Roberto Cannata, Vincenzo Catoni, and Paolo Zampetti. "Two-dimensional hypercomplex numbers and related trigonometries and geometries." Advances in Applied Clifford Algebras 14, no. 1 (2004): 47-68.
 * Rochon, Dominic, and Sebastien Tremblay (2005). "Bicomplex quantum mechanics: II. The Hilbert space." arxiv:quant-ph/0510203
 * Khrennikov, Andrei, and Gavriel Segre (2005). "An introduction to hyperbolic analysis." arxiv:math-ph/0507053
 * Catoni, Francesco, Roberto Cannata, Vincenzo Catoni, and Paolo Zampetti (2005). "Hyperbolic trigonometry in two-dimensional space-time geometry." arXiv:math-ph/0508011
 * Catoni, Francesco, Roberto Cannata, Vincenzo Catoni, and Paolo Zampetti (2005). "Lorentz surfaces with constant curvature and their physical interpretation." arXiv:math-ph/0508012
 * Yamaleev, Robert M. (2005). "Complex algebras on n-order polynomials and generalizations of trigonometry, oscillator model and Hamilton dynamics." Advances in Applied Clifford Algebras 15: 123-150. doi:10.1007/s00006-016-0680-z
 * Ulrych, S. (2005). Relativistic quantum physics with hyperbolic numbers. Physics Letters B, 625(3-4), 313–323. doi:10.1016/j.physletb.2005.08.072
 * Ulrych, S. (2006). "Gravitoelectromagnetism in a complex Clifford algebra." Physics Letters B 633, no. 4-5: 631-635. doi:10.1016/j.physletb.2005.12.050
 * da Rocha, Roldao, and Jayme Vaz Jr (2006). "Extended Grassmann and Clifford algebras." arxiv:math-ph/0603050
 * Boccaletti, Dino, Francesco Catoni, and Vincenzo Catoni (2007). "Space-time trigonometry and formalization of the “Twin Paradox” for uniform and accelerated motions." Advances in applied Clifford algebras 17: 1-22.
 * Ulrych, S. (2008). "Representations of Clifford algebras with hyperbolic numbers." Advances in Applied Clifford Algebras 18, no. 1: 93-114. arxiv:0707.3981 doi:10.1007/s00006-007-0057-4
 * Yüce, S., & Kuruoğlu, N. (2008). One-Parameter Plane Hyperbolic Motions. Advances in Applied Clifford Algebras, 18(2), 279–285. doi:10.1007/s00006-008-0065-z
 * Pogorui, A. A., Rodríguez-Dagnino, R. M., & Rodríguez-Said, R. D. (2008). On the set of zeros of bihyperbolic polynomials. Complex Variables and Elliptic Equations, 53(7), 685–690. doi:10.1080/17476930801973014
 * Demir, Süleyman, Murat Tanışlı, and Nuray Candemir (2010). "Hyperbolic quaternion formulation of electromagnetism." Advances in Applied Clifford Algebras 20: 547-563. doi:10.1007/s00006-010-0209-9
 * Ersoy, Soley, and Mahmut Akyigit (2011). "One-parameter homothetic motion in the hyperbolic plane and Euler-Savary formula." Advances in Applied Clifford Algebras 21, no. 2: 297-313. doi:10.1007/s00006-010-0255-3
 * Hitzer, Eckhard (2013). "Non-constant bounded holomorphic functions of hyperbolic numbers-Candidates for hyperbolic activation functions." arxiv:1306.1653.
 * Sobczyk, Garret (2013). "Complex and hyperbolic numbers." New Foundations in Mathematics: The Geometric Concept of Number: 23-42.
 * Fernández-Guasti, Manuel, and Felipe Zaldívar (2013). "A hyperbolic non distributive algebra in 1+ 2 dimensions." Advances in Applied Clifford Algebras 23, no. 3: 639-656. doi:10.1007/s00006-013-0386-4
 * Trzaska, Zdzisław (2014). "Dynamical processes in sequential-bipolar pulse sources supplying nonlinear loads." Przegląd Elektrotechniczny 90, no. 3: 147-152. http://pe.org.pl/articles/2014/3/32.pdf
 * Trzaska, Z. (2014). "Nonsmooth analysis of the pulse pressured infusion fluid flow." Nonlinear Dynamics 78, no. 1: 525-540. doi:10.1007/s11071-014-1458-2
 * Balci, Yakup, Tulay Erisir, and Mehmet Ali Gungor (2015). "Hyperbolic spinor Darboux equations of spacelike curves in Minkowski 3-space." Journal of the Chungcheong Mathematical Society 28, no. 4: 525-535. https://koreascience.kr/article/JAKO201510659890753.pdf
 * Halldorsson, Hoeskuldur P. "Self-similar solutions to the mean curvature flow in the Minkowski plane ℝ1, 1." Journal für die reine und angewandte Mathematik (Crelles Journal) 2015, no. 704 (2015): 209-243. doi:10.1515/crelle-2013-0054
 * Gargoubi, H., & Kossentini, S. (2016). f-Algebra Structure on Hyperbolic Numbers. Advances in Applied Clifford Algebras, 26(4), 1211–1233. doi:10.1007/s00006-016-0644-3
 * Erdoğdu, Melek, and Mustafa Özdemir (2016). "Matrices over hyperbolic split quaternions." Filomat 30, no. 4: 913-920. jstor:24898658
 * Kumar, Romesh, and Kailash Sharma (2017). "Hyperbolic valued measures and Fundamental law of probability." Global Journal of Pure and Applied Mathematics 13, no. 10: 7163-7177. http://www.ripublication.com/gjpam17/gjpamv13n10_15.pdf
 * Hitzer, Eckhard, and Stephen J. Sangwine (2019). "Construction of Multivector Inverse for Clifford Algebras Over 2 m+ 1 2 m+ 1-Dimensional Vector Spaces from Multivector Inverse for Clifford Algebras Over 2 m-Dimensional Vector Spaces." Advances in Applied Clifford Algebras 29: 1-22. doi:10.1007/s00006-019-0942-7
 * Blankers, Vance, Tristan Rendfrey, Aaron Shukert, and Patrick D. Shipman (2019). "Julia and Mandelbrot sets for dynamics over the hyperbolic numbers." Fractal and fractional 3, no. 1: 6. https://www.mdpi.com/415016
 * Alagöz, Yasemin, and Gözde Özyurt. "Real and hyperbolic matrices of split semi quaternions." Advances in Applied Clifford Algebras 29 (2019): 1-20. doi:10.1007/s00006-019-0973-0
 * Kulyabov, D. S., A. V. Korolkova, and M. N. Gevorkyan (2020). "Hyperbolic numbers as Einstein numbers." In Journal of Physics: Conference Series, vol. 1557, no. 1, p. 012027. IOP Publishing
 * Saini, Heera, Aditi Sharma, and Romesh Kumar (2020). "Some fundamental theorems of functional analysis with bicomplex and hyperbolic scalars." Advances in Applied Clifford Algebras 30: 1-23.
 * Vieira, Guilherme, and Marcos Eduardo Valle (2020). "Extreme Learning Machines on Cayley-Dickson Algebra Applied for Color Image Auto-Encoding." In 2020 International Joint Conference on Neural Networks (IJCNN), pp. 1-8. IEEE. https://ieeexplore.ieee.org/abstract/document/9207495/
 * Soykan, Yuksel, and Melih Göcen (2020). "Properties of hyperbolic generalized Pell numbers." Notes on Number Theory and Discrete Mathematics 26, no. 4: 136-153. https://www.nntdm.net/papers/nntdm-26/NNTDM-26-4-136-153.pdf
 * Golberg, A., and M. E. Luna‐Elizarrarás (2020). "Hyperbolic conformality in multidimensional hyperbolic spaces." Mathematical Methods in the Applied Sciences. doi:10.1002/mma.7109
 * Luna–Elizarrarás, Maria Elena, Marco Panza, Michael Shapiro, and Daniele Carlo Struppa (2020). "Geometry and identity theorems for bicomplex functions and functions of a hyperbolic variable." Milan Journal of Mathematics 88, no. 1: 247-261.
 * Petoukhov, Sergey V. (2020). "Hyperbolic Numbers, Genetics and Musicology." In Advances in Artificial Systems for Medicine and Education III, pp. 195-207. Springer International Publishing. doi:10.1007/978-3-030-39162-1_18
 * Devald, Davor, and Željka Milin Šipuš (2021). "Weierstrass representation for lightlike surfaces in Lorentz-Minkowski 3-space." Journal of Geometry and Physics 166: 104252. doi:10.1016/j.geomphys.2021.104252
 * Soykan, Yüksel (2021). "On hyperbolic numbers with generalized Fibonacci numbers components." Communications in Mathematics and Applications 12, no. 4: 987.
 * Dehdashti, Shahram, Alireza Shahsafi, Bin Zheng, Lian Shen, Zuojia Wang, Rongrong Zhu, Huanyang Chen, and Hongsheng Chen (2021). "Conformal hyperbolic optics." Physical Review Research 3, no. 3: 033281. doi:10.1103/PhysRevResearch.3.033281
 * Petoukhov, Sergey V. (2021). "Modeling inherited physiological structures based on hyperbolic numbers." BioSystems 199: 104285. doi:10.1016/j.biosystems.2020.104285
 * Tellez-Sanchez, G. Y., and Juan Bory-Reyes (2021). "Extensions of the shannon entropy and the chaos game algorithm to hyperbolic numbers plane." Fractals 29, no. 01: 2150013. doi:10.1142/S0218348X21500134
 * Kleine, Vitor G., Ardeshir Hanifi, and Dan S. Henningson (2022). "Stability of two-dimensional potential flows using bicomplex numbers." Proceedings of the Royal Society A 478, no. 2262: 20220165.
 * Korolkova, Anna V., Migran N. Gevorkyan, and Dmitry S. Kulyabov (2023). "Implementation of hyperbolic complex numbers in Julia language." arXiv:2301.01707

double complex numbers:
 * Gürses, Nurten, Mücahit Akbiyik, and Salim Yüce (2016). "One-Parameter Homothetic Motions and Euler-Savary Formula in Generalized Complex Number Plane CJ." Advances in Applied Clifford Algebras 26: 115-136. doi:10.1007/s00006-015-0598-x

double numbers:
 * I. M. Yaglom (1968), Complex numbers in geometry, Academic Press
 * Rooney, Joe (1978). "On the three types of complex number and planar transformations." Environment and Planning B: Planning and Design 5, no. 1: 89-99. https://journals.sagepub.com/doi/pdf/10.1068/b050089
 * Fernández Sanjuan, M. A. (1984). "Group contraction and the nine Cayley-Klein geometries." International journal of theoretical physics 23: 1-14. https://link.springer.com/article/10.1007/BF02080668
 * McCarthy, J. M. (1986). "The generalization of line trajectories in spatial kinematics to trajectories of great circles on a hypersphere". J. Mech., Trans., and Automation. Mar 1986, 108(1): 60-64. doi:10.1115/1.3260785
 * Hazewinkle, M. (1994) "Double and dual numbers", Encyclopaedia of Mathematics, Soviet/AMS/Kluwer, Dordrect.
 * Felsberg, Michael, Thomas Bülow, and Gerald Sommer (2001). "Commutative hypercomplex Fourier transforms of multidimensional signals." Geometric computing with Clifford algebras: theoretical foundations and applications in computer vision and robotics: 209-229. 10.1007/978-3-662-04621-0_9
 * Gadea, P. M., Grifone, J., & Muňoz Masqué, J. (2003). "Manifolds Modelled Over Free Modules Over the Double Numbers". Acta Mathematica Hungarica, 100(3), 187–203. doi:10.1023/a:1025037325005
 * Harkin, A. A., & Harkin, J. B. (2004). Geometry of Generalized Complex Numbers. Mathematics Magazine, 77(2), 118–129. doi:10.1080/0025570x.2004.11953236 http://www.courses.physics.helsinki.fi/fys/tilaII/files/Generalized_Complex_Numbers.pdf
 * Alfsmann, Daniel (2006). "On families of 2 N-dimensional hypercomplex algebras suitable for digital signal processing." In 2006 14th European Signal Processing Conference, pp. 1-4. IEEE.
 * Silagadze, Z. K. (2007). "Relativity without tears." arXiv:0708.0929
 * Kisil, Vladimir V. (2012). "Erlangen program at large: an overview." Advances in applied analysis: 1-94. https://www.emis.de/journals/SIGMA/2010/076/
 * Kisil, Vladimir V. (2010). "Erlangen program at large-1: geometry of invariants." SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 6: 076. https://www.emis.de/journals/SIGMA/2010/076/
 * Kisil, Vladimir V. (2007). "Erlangen Program at Large--2: Inventing a wheel. The parabolic one." arxiv:0707.4024
 * DenHartigh, Kyle, and Rachel Flim (2011). "Liouville theorems in the Dual and Double Planes." Rose-Hulman Undergraduate Mathematics Journal 12, no. 2: 4. https://scholar.rose-hulman.edu/rhumj/vol12/iss2/4/
 * Kisil, Vladimir V. (2012). Geometry Of Mobius Transformations: Elliptic, Parabolic And Hyperbolic Actions Of SL2(R). Imperial College Press.
 * Kisil, Vladimir V. (2012). "Induced representations and hypercomplex numbers." Advances in Applied Clifford Algebras 23, no. 2: 417-440. https://link.springer.com/article/10.1007/s00006-012-0373-1
 * Brewer, S. (2012). Projective Cross-ratio on Hypercomplex Numbers. Advances in Applied Clifford Algebras, 23(1), 1–14. doi:10.1007/s00006-012-0335-7
 * Blom, Conrad, Timothy DeVries, Andrew Hayes, and Daiwei Zhang (2013). "Analytic Extension and Conformal Mapping in the Dual and the Double Planes." Rose-Hulman Undergraduate Mathematics Journal 14, no. 2: 9. https://scholar.rose-hulman.edu/rhumj/vol14/iss2/9/
 * Yefremov, Alexander P. (2014). "Physical theories in hypercomplex geometric description." International Journal of Geometric Methods in Modern Physics 11, no. 06: 1450062.
 * Rooney, J. (2014). "Generalised complex numbers in mechanics." In Advances on Theory and Practice of Robots and Manipulators: Proceedings of Romansy 2014 XX CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators, pp. 55-62. Springer International Publishing. doi:10.1007/978-3-319-07058-2_7
 * Pavlov, Dmitry, and Sergey Kokarev (2015). "The hyperbolic field theory on the plane of double variable." arXiv:1502.06985
 * Mustafa, Khawlah A. (2018). "The groups of two by two matrices in double and dual numbers, and associated Möbius transformations." Advances in Applied Clifford Algebras 28: 1-25. https://link.springer.com/article/10.1007/s00006-018-0910-7
 * Öztürk, İskender, and Mustafa Özdemir (2020). "Similarity of hybrid numbers." Mathematical Methods in the Applied Sciences 43, no. 15: 8867-8881. https://onlinelibrary.wiley.com/doi/abs/10.1002/mma.6580
 * Biswas, D., & Dutta, S. (2020). Möbius Action of SL(2; R) on Different Homogeneous Spaces. Proceedings of the National Academy of Sciences, India Section A: Physical Sciences. doi:10.1007/s40010-020-00673-1
 * Petoukhov, Sergey V. (2021). "Relations of the Genetic Code with Algebraic Codes and Hypercomplex Double Numbers." In Advances in Computer Science for Engineering and Education III 3, pp. 558-570. Springer International Publishing. doi:10.1007/978-3-030-55506-1_50
 * Gutin, Ran (2022). "An analogue of the relationship between SVD and pseudoinverse over double-complex matrices." Linear and Multilinear Algebra: 1-19. doi:10.1080/03081087.2022.2076798