Smarandache Geometries

An axiom is said smarandachely denied if in the same space the axiom behaves differently (i.e., validated and invalided; or only invalidated but in at least two distinct ways). Therefore, we say that an axiom is partially negated, or there is a degree of negation of an axiom.

A Smarandache Geometry is a geometry which has at least one smarandachely denied axiom (1969).

Thus, as a particular case, Euclidean, Lobachevsky-Bolyai-Gauss, and Riemannian geometries may be united altogether, in the same space, by some Smarandache geometries. These last geometries can be partially Euclidean and partially Non-Euclidean.

(i) For the first case, when the axiom is partially validated (true) and partially invalidated (false), the Smarandache Geometries are particular cases of the NeutroGeometry.

(ii) For the second case, when the axiom is only invalidated (100% false), then the Smarandache Geometry are particular cases of the AntiGeometry.

The most important contribution of Smarandache Geometries was the introduction of the degree of negation of an axiom (and more general the degree of negation of a theorem, lemma, property, algorithm, scientific or humanistic proposition, etc.) which works somehow like the negation in fuzzy logic (with a degree of truth, and a degree of falsehood) or more general like the negation in neutrosophic logic (with a degree of truth, a degree of falsehood, and a degree of neutrality (neither true nor false, but unknown, ambiguous, indeterminate) [not only Euclid geometrical axioms, but any scientific or humanistic proposition in any field] or partial negation of an axiom (and, in general, partial negation of a scientific or humanistic proposition in any field).

These geometries connect many geometrical spaces with different structures into a heterogeneous multispace with multistructure.

In general, a rule R ∈ R in a system (Σ; R) is said to be Smarandachely denied if it behaves in at least two different ways within the same set Σ, i.e. validated and invalided, or only invalided but in multiple distinct ways.

A Smarandache system (Σ; R) is a system which has at least one Smarandachely denied rule in R.

In particular, a Smarandache geometry is such a geometry in which there is at least one Smarandachely denied rule, and a Smarandache manifold (M;A) is an n-dimensional manifold M that supports a Smarandache geometry.

In a Smarandache geometry, the points, lines, planes, spaces, triangles, ... are respectively called s-points, s-lines, s-planes, s-spaces, s-triangles, ... in order to distinguish them from those in classical geometry.

Howard Iseri constructed the Smarandache 2-manifolds by using equilateral triangular disks on Euclidean plane R². Such manifold came true by paper models in R³for elliptic, Euclidean and hyperbolic cases. It should be noted that a more general Smarandache n-manifold, i.e. combinatorial manifold and a differential theory on such manifold were constructed by Linfan Mao.

A curve and a surface in a Smarandache Geometry are called a Smarandache Curve and Smarandache Surface respectively. They actually are curves and surfaces of hybrid geometrical structures.

Nearly all geometries, such as pseudo-manifold geometries, Finsler geometry, combinatorial Finsler geometries, Riemann geometry, combinatorial Riemannian geometries, Weyl geometry, Kahler geometry are particular cases of Smarandache geometries.

[Prof. Dr. Linfan Mao, Chinese Academy of Sciences, Beijing, P. R. China, 2005-2023]

Particular cases of Smarandache Geometries:

paradoxist, non-geometry, counter-projective, anti-geometry

Books:

Linfan Mao (2017). Let’s Flying by Wing - Mathematical Combinatorics & Smarandache Multi-Spaces (让我们插上翅膀飞翔 -- 数学组合与Smarandache重叠空间). Chinese Branch Xiquan House, 352 p.

Hu Chang-Wei (2012). Vacuum, Space-Time, Matter and the Models of Smarandache Geometry (真空、时空、物质和). Educational Publishers, 112 p.

Linfan Mao (2011). Combinatorial Geometry with Applications to Field Theory. Education Publisher, 484 p.

Yanpei Liu (2010). Introductory Map Theory. Kapa & Omega, 502 p.

Yuhua Fu, Linfan Mao, Mihaly Bencze (ed) (2007). Scientific Elements - Applications to Mathematics, Physics, and Other Sciences. International book series, Vol. 1. ProQuest Information & Learning, 200 p.

Linfan Mao (2006). Smarandache Geometries & Map Theory with Applications (I). Chinese Branch Xiquan House, 200 p.

Linfan Mao (2006). Smarandache Multi-Space Theory. Hexis, 274 p.

Linfan Mao (2005). Automorphism Groups of Maps, Surfaces and Smarandache Geometries. American Research Press, 114 p.

Howard Iseri (2002). Smarandache Manifolds. American Research Press, 96 p.

Articles:

F. Smarandache (2011). Degree of Negation of Euclid's Fifth Postulate. University of New Mexico, 6 p.

Linfan Mao (2007). A generalization of Stokes theorem on combinatorial manifolds. 16 p.; http://lanl.arxiv.org/abs/math/0703400v1; http://xxx.lanl.gov/pdf/math/0703400v1.

Linfan Mao (2006). Combinatorial Speculations and the Combinatorial Conjecture for Mathematics. 19 p.; http://lanl.arxiv.org/abs/math/0606702v2; http://xxx.lanl.gov/pdf/math/0606702v2.

Linfan Mao (2006). Pseudo-Manifold Geometries with Applications. 15 p.; http://lanl.arxiv.org/abs/math/0610307v1; http://xxx.lanl.gov/pdf/math/0610307v1.

Linfan Mao (2006). Geometrical Theory on Combinatorial Manifolds. 37 p.; http://lanl.arxiv.org/abs/math/0612760v1; http://xxx.lanl.gov/pdf/math/0612760v1.

Linfan Mao (2005). A new view of combinatorial maps by Smarandache's notion. 19 p.; http://lanl.arxiv.org/abs/math/0506232v1; http://xxx.lanl.gov/pdf/math/0506232v1

Linfan Mao (2005). Parallel bundles in planar map geometries. 16 p.; http://lanl.arxiv.org/abs/math/0506386v1; http://xxx.lanl.gov/pdf/math/0506386v1.

L. Kuciuk, M. Antholy (2005). An Introduction to the Smarandache Geometries. JP Journal of Geometry & Topology, 5(1), 77-81.

S. Bhattacharya (2005). A Model to a Smarandache Geometry. Alaska Pacific University, presentation.

Ovidiu Sandru (2004). Un model simplu de geometrie Smarandache construit exclusiv cu elemente de geometrie euclidiana. Universitatea Politehnica Bucharest, Romania, 3 p.

Howard Iseri (2003). A Classification of s-Lines in a Closed s-Manifold. Mansfield University, 3 p.

Howard Iseri (2003). Partially Paradoxist Smarandache Geometries. Mansfield University, 8 p.

Roberto Torretti (2002). An Economics Model for the Smarandache Anti-Geometry. Universidad de Chile, 12 p.

Clifford Singer (2001). Engineering A Visual Field. New York, presentation.

Smarandache Curves and Surfaces

Articles

Süleyman Şenyurt, Kebire Hilal Ayvacı, Davut Canli, (R2026) Special Smarandache Ruled Surfaces According to Flc Frame in E^3, Applications and Applied Mathematics: An International Journal (AAM), Vol. 18, Iss. 1, Article 16, 2023.

Emad Solouma, Ibrahim Al-Dayel, Meraj Ali Khan and Mohamed Abdelkawy, Investigation of Special Type-Π Smarandache Ruled Surfaces Due to Rotation Minimizing Darboux Frame in E³, Symmetry 2023, 15(12), 2207.

Stuti Tamta and Ram Shankar Gupta, Pointwise 1-Type Gauss Map of Developable Smarandache Ruled Surfaces in E^3, Facta Universitatis (NIS), Ser. Math. Inform. Vol. 38, No. 1, 741-759, 2023.

Süleyman ŞENYURT and Davut CANLI, ON THE TANGENT INDICATRIX OF SPECIAL VIVIANI’S
CURVE AND ITS CORRESPONDING SMARANDACHE CURVES ACCORDING TO SABBAN FRAME, 10th International Baskent Congress on Physical, Engineering, and Applied Sciences, ONLINE & IN-PERSON
PARTICIPATION ZOOM & ANKARA, TURKIYE, pp.125-136, OCTOBER 28-30, 2023.

Soukaina Ouarab: NC-Smarandache Ruled Surface and NW-Smarandache Ruled Surface according to Alternative Moving Frame in E3. Hindawi: Journal of Mathematics, Volume 2021, Article ID 9951434, 6 p.; DOI: 10.1155/2021/9951434

Zuhal Kucukarslan Yuzbasi: On Characterizations of Curves in The Galilean Plane. Palestine Journal of Mathematics, Vol. 10(1) (2021), 308–311, 4 p.

Emad Solouma: Equiform Spacelike Smarandache Curves of Anti-Eqiform Salkowski Curve According to Equiform Frame. International Journal of Mathematical Analysis, Vol. 15, 2021, no. 1, 43-59, 17 p.; DOI: 10.12988/ijma.2021.912141

Shankar Lal: Parallel Transport Frame of Smarandache Curves in Euclidean Space. J. Mountain Res., Vol. 16(1), (2021), 225-233, 9 p.; DOI: 10.51220/jmr.v16i1.23

Mustafa Altin, Ahmet Kazan, H.Bayram Karadag: Ruled and Rotational Surfaces Generated by Non-Null Curves with Zero Weighted Curvature. International Electronic Journal of Geometry, Volume 13, no. 2 11–29 (2020), 19 p.; DOI: 10.36890/IEJG.599817

Emad Solouma, (R1977) On Geometry of Equiform Smarandache Ruled Surfaces via Equiform Frame in Minkowski 3-Space, Applications and Applied Mathematics: An International Journal (AAM), Vol. 18, Iss. 1, article 7, pp.1-14, 2023,

Tanju Kahraman, Hasan Huseyin Ugurlu: Smarandache Curves of a Spacelike Curve lying on Unit dual Lorentzian Sphere. CBU J. of Sci., Volume 11, Issue 2, p 93-105, 13 p.

Kahraman Esen Ozen, Murat Tosun: Trajectories Generated by Special Smarandache Curves According to Positional Adapted Frame. 12 p.; www.researchgate.net/publication/348759696

F. Almaz, & M.A. Kulahci. (2019). A Note on Special Smarandache Curves in The Null Cone Q³. Acta Universitatis Apulensis, No. 56/2018, pp. 111-124, http://doi.org/10.5281/zenodo.2987357

M. Khalifa Saad and R. A. Abdel-Baky, On Ruled Surfaces According to Quasi-Frame in Euclidean 3-Space [on Smarandache curves], Aust. J. Math. Anal. Appl. Vol. 17 (2020), No. 1, Art. 11, 16 pp.

H.S. Abdel-Aziz, & M. Khalifa Saad. (2017). Computation of Smarandache curves according to Darboux frame in Minkowski 3-space. Journal of the Egyptian Mathematical Society 25 (2017) 382–390. http://doi.org/10.5281/zenodo.2987485

Tanju Kahraman, Mehmet Onder, & H. Huseyin Ugurlu. (2019). Dual Smarandache Curves and Smarandache Ruled Surfaces. http://doi.org/10.5281/zenodo.2987568

Süleyman Senyurt, Yasin Altun, Ceyda Cevahir, & Huseyin Kocayigit. (2019). On The Sabban Frame Belonging To Involute-Evolute Curves. http://doi.org/10.5281/zenodo.2989788

Mervat Elzawy. (2017). Smarandache curves in Euclidean 4- space E⁴. Journal of the Egyptian Mathematical Society 25 / 2017, 268–271. http://doi.org/10.5281/zenodo.2989884

Suleyman Senyurt, & Abdussamet Calskan. (2015). Smarandache Curves in Terms of Sabban Frame of Spherical Indicatrix Curves. Gen. Math. Notes, Vol. 31, No. 2, December 2015, pp.1-15, http://doi.org/10.5281/zenodo.2990072

M. Elzawy, & S. Mosa. (2017). Smarandache curves in the Galilean 4-space G₄. Journal of the Egyptian Mathematical Society 25 / 2017, 53–56 http://doi.org/10.5281/zenodo.2990158

Suleyman Senyurt, Yasin Altun, Ceyda Cevahir, & Huseyin Kocayigit. (2019). Some Special Curves Belonging to Mannheim Curves Pair. http://doi.org/10.5281/zenodo.2990510

M. Khalifa Saad. (2016). Spacelike and timelike admissible Smarandache curves in pseudo-Galilean space. Journal of the Egyptian Mathematical Society 24 , 416–423. http://doi.org/10.5281/zenodo.2990574

E.M. Solouma. (2019). Special equiform Smarandache curves in Minkowski space-time. Journal of the Egyptian Mathematical Society 25 (2017) 319–325, http://doi.org/10.5281/zenodo.2990660

Elham Mehdi-Nezhad, & Amir M. Rahimi. (2010). The Smarandache Vertices of The Comaximal Graph of A Commutative Ring. http://doi.org/10.5281/zenodo.2990970

Elham Mehdi-Nezhad, Amir M. Rahimi: The Smarandache Vertices of the Comaximal Graph of a Commutative Ring. 12 p. DOI: https://doi.org/10.5281/zenodo.1419756

Mihriban Kulahci, Fatma Almaz: Assesment of Smarandache Curves in the Null Cone Q². 12 p. http://doi.org/10.5281/zenodo.1412498

Esra Betul Koc Ozturk, Ufuk Ozturk, Kazim Elarslan, Emilija Nesovic: On Pseudohyperbolical Smarandache Curves in Minkowski 3-Space. In International Journal of Mathematics and Mathematical Sciences, 2013, 8 pages. http://doi.org/10.5281/zenodo.1413399

H. S. Abdel-Aziz, & M. Khalifa Saad: On Special Curves According to Darboux Frame in the Three Dimensional Lorentz Space. CMC, vol.54, no.3, pp.229-249, 2018. http://doi.org/10.5281/zenodo.1413401

Tanju Kahraman: Smarandache Curves of Null Quaternionic Curves in Minkowski 3-space. In MANAS Journal of Engineering (MJEN), Volume 6, Issue 1, 2018, pp. 1-6. http://doi.org/10.5281/zenodo.1413905

H. S. Abdel-Aziz, & M. Khalifa Saad: On Special Curves According to Darboux Frame in the Three Dimensional Lorentz Space. CMC, vol.54, no.3, pp.229-249, 2018. http://doi.org/10.5281/zenodo.1413401

Tevk Sahin, Merve Okur: Special Smarandache Curves with Respect to Darboux Frame in Galilean 3-Space. 2018, 15 pages. http://doi.org/10.5281/zenodo.1413956

Roberto Torretti: A model for the Smarandache anti-geometry. In Int. Journal of Social Economics, vol. 29, nr. 11, 2002, 886-896. http://doi.org/10.5281/zenodo.1412417

Gulnur Saffak Atalay: Surfaces family with a common Mannheim geodesic curve. In Journal of Applied Mathematics and Computation (JAMC), 2018, 2(4), pp. 155-165. http://doi.org/10.5281/zenodo.1413970

V. Ramachandran: (1,N)-Arithmetic Labelling of Ladder and Subdivision of Ladder, International J. Math. Combin. Vol. 2 (2018), pp. 114-121. DOI: http://doi.org/10.5281/zenodo.1418773

R. Ponraj and M. Maria Adaickalam: 3-Difference Cordial Labeling of Corona Related Graphs, International J. Math. Combin. Vol. 2 (2018), pp. 122-128. DOI: http://doi.org/10.5281/zenodo.1418782

R. Ponraj, K. Annathurai, R. Kala: 4-Remainder Cordial Labeling of Some Graphs, International J. Math. Combin. Vol. 1 (2018), pp. 138-145. DOI: http://doi.org/10.5281/zenodo.1418786

B. Basavanagoud, Sujata Timmanaikar: Accurate Independent Domination in Graphs, International J. Math. Combin. Vol. 2 (2018), pp. 87-96. DOI: http://doi.org/10.5281/zenodo.1418792

Rajesh Kumar T.J., Mathew Varkey T.K.: Adjacency Matrices of Some Directional Paths and Stars, International J. Math. Combin. Vol. 1 (2018), pp. 90-96. DOI: http://doi.org/10.5281/zenodo.1418808

M. Subramanian, T. Subramanian: A Study on Equitable Triple Connected Domination Number of a Graph, International J. Math. Combin. Vol. 3 (2017), pp. 116-118. DOI: http://doi.org/10.5281/zenodo.1418828

U M Prajapati, R M Gajjar: Cordiality in the Context of Duplication in Web and Armed Helm, International J. Math. Combin. Vol. 3 (2017), pp. 90-115. DOI: http://doi.org/10.5281/zenodo.1418910

T. Chalapathi, R.V M S S Kiran Kumar: Equal Degree Graphs of Simple Graphs, International J. Math. Combin. Vol. 1 (2018), pp. 127-137. DOI: http://doi.org/10.5281/zenodo.1418906

K. Praveena, M. Venkatachalam: Equitable Coloring on Triple Star Graph Families, International J. Math. Combin. Vol. 2 (2018), pp. 24-32. DOI: http://doi.org/10.5281/zenodo.1418902

K. Muthugurupackiam, S. Ramya: Even Modular Edge Irregularity Strength of Graphs, International J. Math. Combin. Vol. 1 (2018), pp. 75-82. DOI: http://doi.org/10.5281/zenodo.1418898

Linfan Mao: Hilbert Flow Spaces with Operators over Topological Graphs, International J. Math. Combin. Vol. 4 (2017), pp. 19-45. DOI: http://doi.org/10.5281/zenodo.1418874

A. Lourdusamy, Sherry George: Linear Cyclic Snakes as Super Vertex Mean Graphs, International J. Math. Combin. Vol. 1 (2018), pp. 109-126. DOI: http://doi.org/10.5281/zenodo.1418960

P.S.K. Reddy, K.N. Prakasha, Gavirangaiah K.: Minimum Dominating Color Energy of a Graph, International J. Math. Combin. Vol. 3 (2017), pp. 22-31. DOI: http://doi.org/10.5281/zenodo.1418990

Rajendra P., R. Rangarajan: Minimum Equitable Dominating Randic Energy of a Graph, International J. Math. Combin. Vol. 1 (2018), pp. 97-108. DOI: http://doi.org/10.5281/zenodo.1419017

P. S. K. Reddy, K. N. Prakasha, Gavirangaiah K.: Minimum Equitable Dominating Randic Energy of a Graph, International J. Math. Combin. Vol. 3 (2017), pp. 81-89. DOI: http://doi.org/10.5281/zenodo.1419027

T. Deepa, M. Venkatachalam: On r-Dynamic Coloring of the Triple Star Graph Families, International J. Math. Combin. Vol. 2 (2018), pp. 97-113. DOI: http://doi.org/10.5281/zenodo.1419043

Akram Alqesmah, Anwar Alwardi, R. Rangarajan: On the Distance Eccentricity Zagreb Indeices of Graphs, International J. Math. Combin. Vol. 4 (2017), pp. 110-120. DOI: http://doi.org/10.5281/zenodo.1418934

A. Nellai Murugan, P. Iyadurai Selvaraj: Path Related n-Cap Cordial Graphs, International J. Math. Combin. Vol. 3 (2017), pp. 119-124. DOI: http://doi.org/10.5281/zenodo.1419057

Ujwala Deshmukh, Smita A. Bhatavadeka: Primeness of Supersubdivision of Some Graphs, International J. Math. Combin. Vol. 4 (2017), pp. 151-156. DOI: http://doi.org/10.5281/zenodo.1418938

V. Lokesha, P. S. Hemavathi, S. Vijay: Semifull Line (Block) Signed Graphs, International J. Math. Combin. Vol. 2 (2018), pp. 80-86. DOI: http://doi.org/10.5281/zenodo.1419004

Tanju Kahraman, Hasan Huseyin Ugurlu: Smarandache Curves of Curves lying on Lightlike Cone…, International J. Math. Combin. Vol. 3 (2017), pp. 1-9. DOI: http://doi.org/10.5281/zenodo.1419069

R. Ponraj, Rajpal Singh, R. Kala: Some New Families of 4-Prime Cordial Graphs, International J. Math. Combin. Vol. 3 (2017), pp. 125-135. DOI: http://doi.org/10.5281/zenodo.1419037

M. H. Akhbari, F. Movahedi, S. V. R. Kulli: Some Parameters of Domination on the Neighborhood Graph, International J. Math. Combin. Vol. 4 (2017), pp. 138-150. DOI: http://doi.org/10.5281/zenodo.1418998

Samir K. Vaidya, Raksha N. Mehta: Strong Domination Number of Some Cycle Related Graphs, International J. Math. Combin. Vol. 3 (2017), pp. 72-80. DOI: http://doi.org/10.5281/zenodo.1419079

Ahmed M. Naji and Soner Nandappa D.: The k-Distance Degree Index of Corona, Neighborhood Corona Products and Join of Graphs, International J. Math. Combin. Vol. 4 (2017), pp. 91-102. DOI: http://doi.org/10.5281/zenodo.1419023

Suleyman Senyurt, Selin Sivas: An Application of Smarandache Curve. In Ordu Univ. Bil. Tek. Derg., Cilt: 3, Sayi:1, 2013,46-60/Ordu Univ. J. Sci. Tech., Vol. 3, No. 1 (2013), 15 pages.

H.S. Abdel-Aziz, M. Khalifa Saad: Computation of Smarandache curves according to Darboux frame in Minkowski 3-space. In Journal of the Egyptian Mathematical Society, 25 (2017), pp. 382-390, 9 pages.

Mervat Elzawy: Smarandache curves in Euclidean 4-space E⁴. In Journal of the Egyptian Mathematical Society, 25 (2017), pp. 268-271, 4 pages.

M. Elzawy, S. Mosa: Smarandache curves in the Galilean 4-space G₄. In Journal of the Egyptian Mathematical Society, 25 (2017), pp. 53-56, 4 pages.

M. Khalifa Saad: Spacelike and timelike admissible Smarandache curves in pseudo-Galilean space. In Journal of the Egyptian Mathematical Society, 24 (2016), pp. 416-423, 8 pages.

E.M. Solouma: Special equiform Smarandache curves in Minkowski space-time. In Journal of the Egyptian Mathematical Society, 25 (2017), pp. 319-325, 7 pages.

E.M. Solouma: Special timelike Smarandache curves in Minkowski 3-space. Al Imam Mohammad Ibn Saud Islamic University, College of Science, Department of Mathematics and Statistics, KSA, Riyadh, 16 pages.

b−Smarandache m₁m₂ Curves of Biharmontic New Type b−Slant Helices According to Bishop Frame in the Sol Space Sol³, by Talat Korpinar, Essin Turhan. In International J.Math. Combin. Vol. 4 (2012), pp. 33-39. DOI:10.5281/zenodo.825679, https://doi.org/10.5281/zenodo.825678

Isotropic Smarandache Curves in Complex Space C³, by Suha Yilmaz. In International J.Math. Combin. Vol. 4 (2016), 7 pages. DOI:10.5281/zenodo.826790, https://doi.org/10.5281/zenodo.826789

Dual Smarandache Curves and Smarandache Ruled Surfaces, by Tanju Kahraman, Mehmet Onder, H. Huseyin Ugurlu, In Mathematical Sciences and Applications E-Notes, Volume 2 No. 1, pp. 83/98 (2014), 16 pages. DOI:10.5281/zenodo.835438, https://doi.org/10.5281/zenodo.835437

Mannheim Partner Curve a Different View, Suleyman Senyurt, Yasin Altun, Ceyda Cevahir. In International J.Math. Combin. Vol. 2 (2017), pp. 84-91. DOI:10.5281/zenodo.831975, https://doi.org/10.5281/zenodo.831974

Ruled surfaces generated by some special curves in Euclidean 3-Space, by Ahmad T. Ali, Hossam S. Abdel Aziz, Adel H. Sorour. In Journal of the Egyptian Mathematical Society (2013) 21, pp. 285–294. DOI:10.5281/zenodo.835445, https://doi.org/10.5281/zenodo.835444

N*C*-Smarandache Curve of Bertrand Curves Pair According to Frenet Frame, by Suleyman Senyurt, Abdussamet Caliskan, Unzile Celik. In International J.Math. Combin. Vol. 1 (2016), 7 pages. DOI:10.5281/zenodo.815716, https://doi.org/10.5281/zenodo.815715

N*C*-Smarandache Curves of Mannheim Curve Couple According to Frenet Frame, by Suleyman Senyurt, Abdussamet Caliskan. In International J.Math. Combin. Vol. 1 (2015), 13 pages. DOI:10.5281/zenodo.815112, https://doi.org/10.5281/zenodo.815111

New type surfaces in terms of B-Smarandache Curves in Sol³, by Talat Korpinar. In Acta Scientiarum. Technology, Maringá, v. 37, n. 3, July-Sept. 2015, pp. 389-393. DOI:10.5281/zenodo.835440, https://doi.org/10.5281/zenodo.835439

On Pseudospherical Smarandache Curves in Minkowski 3-Space, by Esra Betul Koc Ozturk, Ufuk Ozturk, Kazim Ilarslan, Emilija Nesovic. In Journal of Applied Mathematics, Volume 2014, Article ID 404521, 14 pages. DOI:10.5281/zenodo.835443, https://doi.org/10.5281/zenodo.835442

Smarandache Curves According to Bishop Frame in Euclidean 3-Space, by Muhammed Cetin, Yilmaz Tuncer, Murat Kemal Karacan. arXiv:1106.3202v1 [math.GM] 16 Jun 2011, 19 pages. DOI:10.5281/zenodo.835447, https://doi.org/10.5281/zenodo.835446

Smarandache Curves According to Sabban Frame on S², by Kemal Taskopru, Murat Tosun. arXiv:1206.6229v3 [math.DG] 20 Jul 2012, 8 pages. DOI:10.5281/zenodo.835452, https://doi.org/10.5281/zenodo.835451

Smarandache Curves and Applications According to Type-2 Bishop Frame in Euclidean 3-Space, by Suha Yilmaz, Umit Ziya Savcı. In International J.Math. Combin. Vol. 2 (2016), 15 pages. DOI:10.5281/zenodo.822231, https://doi.org/10.5281/zenodo.822230

Smarandache Curves and Spherical Indicatrices in the Galilean 3-Space, by H.S. Abdel-Aziz, M. Khalifa Saad. arXiv:1501.05245v1 [math.DG] 21 Jan 2015, 15 pages. DOI:10.5281/zenodo.835456, https://doi.org/10.5281/zenodo.835455

Smarandache Curves in Euclidean 4- space E⁴, by Mervat Elzawy. In Journal of the Egyptian Mathematical Society (2017), 4 pages. DOI:10.5281/zenodo.835458, https://doi.org/10.5281/zenodo.835457

Smarandache Curves in Minkowski Space-time, by Melih Turgut, Suha Yilmaz. In International J.Math. Combin. Vol.3 (2008), pp. 51-55. DOI:10.5281/zenodo.823501, https://doi.org/10.5281/zenodo.823500

Smarandache Curves in Terms of Sabban Frame of Fixed Pole Curve, by Suleyman Senyurt, Abdussamet Caliskan. In Bol. Soc. Paran. Mat. (3s.) v. 34 2 (2016), pp. 53–62. DOI:10.5281/zenodo.835460, https://doi.org/10.5281/zenodo.835459

Smarandache curves in the Galilean 4-space G⁴, by M. Elzawy, S. Mosa. In Journal of the Egyptian Mathematical Society (2016), 4 pages. DOI:10.5281/zenodo.835462, https://doi.org/10.5281/zenodo.835459

Smarandache Curves of a Spacelike Curve According to the Bishop Frame of Type-2, by Yasin Unluturk, Suha Yilmaz. In International J.Math. Combin. Vol. 4 (2016), pp. 29-43. DOI:10.5281/zenodo.826804, https://doi.org/10.5281/zenodo.826803

Smarandache curves of some special curves in the Galilean 3-space, by H. S. Abdel-Aziz, M. Khalifa Saad. arXiv:1501.05245v2 [math.DG] 19 Feb 2015, 11 pages. DOI: 10.5281/zenodo.835464, https://doi.org/10.5281/zenodo.835463

Spacelike Smarandache Curves of Timelike Curves in Anti de Sitter 3-Space, by Mahmut Mak, Hasan Altinbas. In International J.Math. Combin. Vol. 3 (2016), 16 pages. DOI:10.5281/zenodo.825056, https://doi.org/10.5281/zenodo.825055

Special Smarandache Curves in R³₁, by Nurten (Bayrak) Gurses, Ozcan Bektas, Salim Yuce. In Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Volume 65, Number 2, 2016, pp. 143-160. DOI:10.5281/zenodo.835466, https://doi.org/10.5281/zenodo.835465

Special Smarandache Curves According to Bishop Frame in Euclidean Spacetime, by E. M. Solouma, M. M. Wageeda. In International J.Math. Combin. Vol. 1 (2017), 9 pages. DOI:10.5281/zenodo.815770, https://doi.org/10.5281/zenodo.815769

Special Smarandache Curves in the Euclidean Space, by Ahmad T. Ali. In International J.Math. Combin. Vol. 2 (2010), pp. 30-36. DOI:10.5281/zenodo.821048, https://doi.org/10.5281/zenodo.821047

Spherical Images of Special Smarandache Curves in E³, by Vahide Bulut, Ali Caliskan. In International J.Math. Combin. Vol. 3 (2015), pp. 43-54. DOI:10.5281/zenodo.825027, https://doi.org/10.5281/zenodo.825026

The Smarandache Curves on H₀²^,, by Murat Savas, Atakan Tugkan Yakut, Tugba Tamirci. In Gazi University Journal of Science GU J Sci. 29(1) (2016), pp. 69-77. DOI:10.5281/zenodo.835469, https://doi.org/10.5281/zenodo.835468

The Smarandache Curves on S²₁ and Its Duality on H²₀, by Atakan Tulkan Yakut, Murat Savas, Tugba Tamirci. In Journal of Applied Mathematics, Volume 2014, Article ID 193586, 12 pages. DOI:10.5281/zenodo.835471, https://doi.org/10.5281/zenodo.835470

M. Khalifa Saad, R. A. Abdel-Baky (2020). On Ruled Surfaces According to Quasi-Frame in Euclidean 3-Space. Aust. J. Math. Anal. Appl. 17(1), Art. 11, 16 p.

Suleyman Senyurt, Yasin Altun, Ceyda Cevahir, Huseyin Kocayigit (2019). On The Sabban Frame Belonging To Involute-Evolute Curves. Ordu University, 11 p. DOI:10.5281/zenodo.2989788.

Suleyman Senyurt, Yasin Altun, Ceyda Cevahir, Huseyin Kocayigit (2019). Some Special Curves Belonging to Mannheim Curves Pair. Ordu University, 10 p. DOI: 10.5281/zenodo.2990510.

F. Almaz, M.A. Kulahci (2018). A Note on Special Smarandache Curves in The Null Cone Q³. Acta Universitatis Apulensis 56, 111-124. DOI: 10.5281/zenodo.2987357.

A. Lourdusamy, Sherry George (2018). Linear Cyclic Snakes as Super Vertex Mean Graphs. International J. Math. Combin. (IJMC) 1, 109-126. DOI: 10.5281/zenodo.1418960.

H. S. Abdel-Aziz, M. Khalifa Saad (2018). On Special Curves According to Darboux Frame in the Three Dimensional Lorentz Space. CMC 54(3), 229-249.

Tanju Kahraman (2018). Smarandache Curves of Null Quaternionic Curves in Minkowski 3-space. MANAS Journal of Engineering (MJEN) 6(1), 6 p. DOI: 10.5281/zenodo.1413905.

Tevk Sahin, Merve Okur (2018). Special Smarandache Curves with Respect to Darboux Frame in Galilean 3-Space. Int. J. Adv. Appl. Math. and Mech. 5(3), 15-26.

Gulnur Saffak Atalay (2018). Surfaces family with a common Mannheim geodesic curve. J. Appl. Math. Comp. (JAMC) 2(4), 155-165.

V. Ramachandran (2018). (1,N)-Arithmetic Labelling of Ladder and Subdivision of Ladder. International J. Math. Combin. (IJMC) 2, 114-121.

R. Ponraj, M. Maria Adaickalam (2018). 3-Difference Cordial Labeling of Corona Related Graphs. International J. Math. Combin. (IJMC) 2, 122-128.

R. Ponraj, K. Annathurai, R. Kala (2018). 4-Remainder Cordial Labeling of Some Graphs. International J. Math. Combin. (IJMC) 1, 138-145. DOI: 10.5281/zenodo.1418786.

B. Basavanagoud, Sujata Timmanaikar (2018). Accurate Independent Domination in Graphs. International J. Math. Combin. (IJMC) 2, 87-96.

Rajesh Kumar T.J., Mathew Varkey T.K. (2018). Adjacency Matrices of Some Directional Paths and Stars. International J. Math. Combin. (IJMC) 1, 90-96. DOI: 10.5281/zenodo.1418808.

T. Chalapathi, R.V M S S Kiran Kumar (2018). Equal Degree Graphs of Simple Graphs. International J. Math. Combin. (IJMC) 1, 127-137.

K. Praveena, M. Venkatachalam (2018). Equitable Coloring on Triple Star Graph Families. International J. Math. Combin. (IJMC) 2, 24-32. DOI: 10.5281/zenodo.1418902.

K. Muthugurupackiam, S. Ramya (2018). Even Modular Edge Irregularity Strength of Graphs. International J. Math. Combin. (IJMC) 1, 75-82.

V. Lokesha, P. S. Hemavathi, S. Vijay (2018). Semifull Line (Block) Signed Graphs. International J. Math. Combin. (IJMC) 2, 80-86.

Rajendra P., R. Rangarajan (2018). Minimum Equitable Dominating Randic Energy of a Graph. International J. Math. Combin. (IJMC) 1, 97-108.

T. Deepa, M. Venkatachalam (2018). On r-Dynamic Coloring of the Triple Star Graph Families. International J. Math. Combin. (IJMC) 2, 97-113.

Tanju Kahraman, Hasan Huseyin Ugurlu (2017). Smarandache Curves of Curves lying on Lightlike Cone. International J. Math. Combin. (IJMC) 3, 1-9.

R. Ponraj, Rajpal Singh, R. Kala (2017). Some New Families of 4-Prime Cordial Graphs. International J. Math. Combin. (IJMC) 3, 125-135.

M. H. Akhbari, F. Movahedi, S. V. R. Kulli (2017). Some Parameters of Domination on the Neighborhood Graph. International J. Math. Combin. (IJMC) 4, 138-150.

Samir K. Vaidya, Raksha N. Mehta (2017). Strong Domination Number of Some Cycle Related Graphs. International J. Math. Combin. (IJMC) 3, 72-80.

Ahmed M. Naji, Soner Nandappa D. (2017). The k-Distance Degree Index of Corona, Neighborhood Corona Products and Join of Graphs. International J. Math. Combin. (IJMC) 4, 91-102.

Suleyman Senyurt, Yasin Altun, Ceyda Cevahir (2017). Mannheim Partner Curve a Different View. International J.Math. Combin. (IJMC) 2, 84-91.

H.S. Abdel-Aziz, M. Khalifa Saad (2017). Computation of Smarandache curves according to Darboux frame in Minkowski 3-space. Journal of the Egyptian Mathematical Society 25, 382-390. DOI: 10.5281/zenodo.2987485.

Mervat Elzawy (2017). Smarandache curves in Euclidean 4-space E⁴. Journal of the Egyptian Mathematical Society 25, 268-271. , DOI: 10.5281/zenodo.2989884.

M. Elzawy, S. Mosa (2017). Smarandache curves in the Galilean 4-space G₄. Journal of the Egyptian Mathematical Society 25, 53-56, DOI: 10.5281/zenodo.2990158.

E.M. Solouma (2017). Special equiform Smarandache curves in Minkowski space-time. Journal of the Egyptian Mathematical Society 25, 319-325. DOI: 10.5281/zenodo.2990660.

Akram Alqesmah, Anwar Alwardi, R. Rangarajan (2017). On the Distance Eccentricity Zagreb Indeices of Graphs. International J. Math. Combin. (IJMC) 4, 110-120. DOI: 10.5281/zenodo.1418934.

A. Nellai Murugan, P. Iyadurai Selvaraj (2017). Path Related n-Cap Cordial Graphs. International J. Math. Combin. (IJMC) 3, 119-124.

Ujwala Deshmukh, Smita A. Bhatavadeka (2017). Primeness of Supersubdivision of Some Graphs. International J. Math. Combin. (IJMC) 4, 151-156.

E. M. Solouma, M. M. Wageeda (2017). Special Smarandache Curves According to Bishop Frame in Euclidean Spacetime. International J.Math. Combin. (IJMC) 1, 1-9.

M. Subramanian, T. Subramanian (2017). A Study on Equitable Triple Connected Domination Number of a Graph. International J. Math. Combin. (IJMC) 3, 116-118.

U. M. Prajapati, R. M. Gajjar (2017). Cordiality in the Context of Duplication in Web and Armed Helm. International J. Math. Combin. (IJMC) 3, 90-115.

Linfan Mao (2017). Hilbert Flow Spaces with Operators over Topological Graphs. International J. Math. Combin. (IJMC) 4, 19-45.

P.S.K. Reddy, K.N. Prakasha, Gavirangaiah K. (2017). Minimum Dominating Color Energy of a Graph. International J. Math. Combin. (IJMC) 3, 22-31.

P. S. K. Reddy, K. N. Prakasha, Gavirangaiah K. (2017). Minimum Equitable Dominating Randic Energy of a Graph. International J. Math. Combin. (IJMC) 3, 81-89.

M. Khalifa Saad (2016). Spacelike and timelike admissible Smarandache curves in pseudo-Galilean space. Journal of the Egyptian Mathematical Society 24, 416-423. DOI: 10.5281/zenodo.2990574.

Suleyman Senyurt, Abdussamet Caliskan (2016). Smarandache Curves in Terms of Sabban Frame of Fixed Pole Curve. Bol. Soc. Paran. Mat. 34(2), 53–62. DOI: 10.5281/zenodo.835460.

Yasin Unluturk, Suha Yilmaz (2016). Smarandache Curves of a Spacelike Curve According to the Bishop Frame of Type-2. International J.Math. Combin. (IJMC) 4, 29-43.

Murat Savas, Atakan Tugkan Yakut, Tugba Tamirci (2016). The Smarandache Curves on H²₀. Gazi University Journal of Science 29(1), 69-77.

Nurten (Bayrak) Gurses, Ozcan Bektas, Salim Yuce (2016). Special Smarandache Curves in R³₁. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 65(2), 143-160. DOI:10.5281/zenodo.835466.

Suleyman Senyurt, Abdussamet Caliskan, Unzile Celik (2016). N*C*-Smarandache Curve of Bertrand Curves Pair According to Frenet Frame. International J.Math. Combin. (IJMC) 1, 1-7.

Suha Yilmaz, Umit Ziya Savci (2016). Smarandache Curves and Applications According to Type-2 Bishop Frame in Euclidean 3-Space. International J.Math. Combin. (IJMC) 2, 1-15. DOI: 10.5281/zenodo.822230.

Mahmut Mak, Hasan Altinbas (2016). Spacelike Smarandache Curves of Timelike Curves in Anti de Sitter 3-Space. International J.Math. Combin. (IJMC) 3, 1-16. DOI: 10.5281/zenodo.825056.

Suha Yilmaz (2016). Isotropic Smarandache Curves in Complex Space C³. International J.Math. Combin. (IJMC) 4, 1-7.

Suleyman Senyurt, Abdussamet Calskan (2015). Smarandache Curves in Terms of Sabban Frame of Spherical Indicatrix Curves. Gen. Math. Notes 31(2), 1-15. DOI: 10.5281/zenodo.2990072.

H. S. Abdel-Aziz, M. Khalifa Saad (2015). Smarandache curves of some special curves in the Galilean 3-space. arXiv:1501.05245v2 [math.DG] 19 Feb 2015, 11 p. DOI: 10.5281/zenodo.835464.

Vahide Bulut, Ali Caliskan (2015). Spherical Images of Special Smarandache Curves in E³. International J.Math. Combin. (IJMC) 3, 43-54.

Suleyman Senyurt, Abdussamet Caliskan (2015). N*C*-Smarandache Curves of Mannheim Curve Couple According to Frenet Frame. International J.Math. Combin. (IJMC) 1, 1-13.

Talat Korpinar (2015). New type surfaces in terms of B-Smarandache Curves in Sol³. Acta Scientiarum. Technology 37(3), 389-393. DOI: 10.5281/zenodo.835440.

H.S. Abdel-Aziz, M. Khalifa Saad (2015). Smarandache Curves and Spherical Indicatrices in the Galilean 3-Space. arXiv:1501.05245v1 [math.DG] 21 Jan 2015, 15 p.

Tanju Kahraman, Mehmet Onder, H. Huseyin Ugurlu (2014). Dual Smarandache Curves and Smarandache Ruled Surfaces. Mathematical Sciences and Applications E-Notes 2(1), 83-98. DOI: 10.5281/zenodo.2987568.

Atakan Tulkan Yakut, Murat Savas, Tugba Tamirci (2014). The Smarandache Curves on S²₁ and Its Duality on H²₀. Journal of Applied Mathematics, Article ID 193586, 12 p.

Esra Betul Koc Ozturk, Ufuk Ozturk, Kazim Ilarslan, Emilija Nesovic (2014). On Pseudospherical Smarandache Curves in Minkowski 3-Space. Journal of Applied Mathematics, Article ID 404521, 14 p. DOI: 10.5281/zenodo.835443.

Esra Betul Koc Ozturk, Ufuk Ozturk, Kazim Ilarslan, Emilija Nesovic (2013). On Pseudohyperbolical Smarandache Curves in Minkowski 3-Space. International Journal of Mathematics and Mathematical Sciences, 8 p. DOI: 10.5281/zenodo.1413399.

Suleyman Senyurt, Selin Sivas (2013). An Application of Smarandache Curve. Ordu Univ. J. Sci. Tech. 3(1), 46-60.

Ahmad T. Ali, Hossam S. Abdel Aziz, Adel H. Sorour (2013). Ruled surfaces generated by some special curves in Euclidean 3-Space. Journal of the Egyptian Mathematical Society 21, 285–294. DOI: 10.5281/zenodo.835445.

Kemal Taskopru, Murat Tosun (2012). Smarandache Curves According to Sabban Frame on S². arXiv:1206.6229v3 [math.DG] 20 Jul 2012, 8 p.

Talat Korpinar, Essin Turhan (2012). b-Smarandache m₁m₂ Curves of Biharmontic New Type b-Slant Helices According to Bishop Frame in the Sol Space Sol³. International J.Math. Combin. (IJMC) 4, 33-39.

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Elham Mehdi-Nezhad, Amir M. Rahimi (2010). The Smarandache Vertices of The Comaximal Graph of A Commutative Ring. Stellenbosch University, 12 p. DOI: 10.5281/zenodo.2990970.

Ahmad T. Ali (2010). Special Smarandache Curves in the Euclidean Space. International J.Math. Combin. (IJMC) 2, 30-36.

Melih Turgut, Suha Yilmaz (2008). Smarandache Curves in Minkowski Space-time. International J.Math. Combin. (IJMC) 3, 51-55. DOI: 10.5281/zenodo.823501.

Roberto Torretti (2002). A model for the Smarandache anti-geometry. Int. Journal of Social Economics 29(11), 886-896. DOI: 10.5281/zenodo.1412417.

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