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AbstractAccording to current information the secrets of the universe can not be solved by using rational numbers because the irrational numbers have been used more than the rational numbers in the creation of universe but the real values of irrational numbers have not been known. The most excellent program of brain is written in left angular gyrus in right handed persons; but unfortunately the nural networks of angular gyrus do not have the understanding ability of irrational numbers so far. And even, angular gyrus lesions can cause the loss of normal mathematical ability that this phemonon is known as The Gerstmann’s Syndrome. Then, we hypothetised that it is impossible to destroy uncertainity problem unless irrational numbers are rationalised. If the neural networks of angular gyrus will more develop by evolution mechanisms, the irrationality problem would be solved in the future? Agulary gyruses analyzed a synoptically by clinical and experimental ways. We show that normal angular gyrus could not solve irrationality and the more developed neural networks of the angular gyrus could succed in the uncertainity problem solving by irrationality. Interestingly, the more clever wild cats have more mathematical ability than domestic cats because wild cats have more glial cells in their bigger angular gyruses than domestic cats. If so, has angular gyrus has not been thoroughly developed to understand irrational numbers? According to current theories, have all of us Gerstmann’s Syndrome?Keywords : Mathematics, Medical |
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References[1] Aydın MD, Malkoc I, Aydin N, Tozoglu EO, Sipal S, Ozmen S, Altınkaynak K: Angular Gırus Ve Glıa Hücrelerının Fraktal Yapısının Matematık Zekası Üzerıne Etkısı: Deneysel Çalışma: TND 32. Bilimsel Kongresi- Türk Nöroşirürji Dergisi-Abstract Book: SS-103: pp:98 [2] Apéry R: Irrationalité de et . Astérisque 61:11-13, 1979 [3] Aydin MD, Aydin N: A Cerebral Hydatid Cyst Case First Presenting with Gerstmanns Syndrome: A Case Report and Literature Review. Turk J Med Sci 33:57-60,2003 [4] Bailey D H, Crandall R E: Random Generators and Normal Numbers." Exper Math 11:527-546, 2002 [5] Bhattacharyya S, Cai X, Klein JP: Dyscalculia, dysgraphia, and left-right confusion from a left posterior peri-insular infarct. Behav Neurol 2014:823591, 2014 [6] Brian Clegg. The Dangerous Ratio: Published in February 2011. Copyright © 1997 - 2018. University of Cambridge. [7] Dehaene S, Brannon EM: Space, time, and number: a Kantian research program.Trends Cogn Sci 12:517-519, 2010 [8] Elmas S,Hızarcı S, The Different Form Of The Golden Ratio Atatürk Üniversitesi Güzel Sanatlar Enstitüsü Dergisi Journal of the Fine Arts Institute (GSED), Sayı / Number 38, Erzurum, 2017, 99-103 [9] Elmas S, Golden Ratio in Plane and Sphere,International Journal Of Academic Research Vol. 7. No. 3. Iss.1. May, 2015 [10] Gary Meisner: Golden Ratio Overview. Celebrity Faces and the Golden Ratio: The real story. June 13, 2015 by Gary Meisner —July 12, 2015 [11] Hardy G H, Wright E M An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979 [12] Livio M: The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books, 2002 [13] Nagell T: Irrational Numbers and Irrationality of the numbers and ." §12-13 in Introduction to Number Theory. New York: Wiley pp: 38-40,1951 [14] Nagell, T. Introduction to Number Theory. New York: Wiley, pp. 38-40, 1951; Brian Clegg: The Dangerous Ratio. February 2011 [15] Niven I M: Irrational Numbers. New York: Wiley, 1956 [16] Pappas T: Irrational Numbers & the Pythagoras Theorem." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp: 98-99, 1989 [17] Pappas, T., 1989. The Joy of Mathematics: Discovering Mathematics All Around You. Vol., Wide World Pub./Tetra. [18] Salvatori R, Albert Einstein's brain. Lancet. 1999 Nov 20;354(9192):1821-2 [19] Wilkey ED, Cutting LE, Price GR: Neuroanatomical correlates of performance in a state-wide test of math achievement. Dev Sci doi: 10.1111/desc.12545, 2017 [20] Wu YT, Chen LC, Lin SL, Chang ST: Gerstmann's syndrome associated with diagnostic cerebral angiography. Brain Inj 272:239-241, 2013 [21] Wilkey, E.D., Cutting, L.E., Price, G.R., 2018. Neuroanatomical correlates of performance in a state-wide test of math achievement. Dev Sci. 21. |
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AbstractIn this study, we examined the Newton-Rapson method from fixed point iterations. With a few examples, we proved the validity of the method again.Keywords : Newton-Rapson method, fixed point iterations and fixed point |
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References[1] Dr. B.D. Pant, S. Kumar, Some Fixed Point Theorems in Menger Spaces and Aplications, Uttarakhand 247712, March 2008. 2 A.McLennan, Advanced Fixed Point Theory for Economics, April 8, 2014. [] 3 S.Elmas, “A Fixed Point Theorem for Triangular Surface Mapping and Functions of Different”, [] International Journal of Environmental & Science Education, 2017, Vol. 12, No. 8, 1743-1749 [3] S.R.Clemens, " Fixed point theorems in Euclidean Geometry," Mathematics Teacher, pp. 324-330, Ap. 1973 [4] Y .Soykan, "Fonksiyonel Analiz", Nobel Yayın Dağıtım Ankara. Turkey, 2008. [5] S.Elmas, "Sabit Nokta İterasyonlarının Yakınsama Hızları,"Atatürk Üniversitesi Fen Bilimleri Enstitüsü Erzurum-Türkiye.2010 [6] S.Elmas, M.Özdemir, Convergence of a General Iterative Scheme for Three Infinite Fammilies of Uniformly Quasi – Lipschitzian Mappings in Convex Metric Spaces”, Advances in Fixed Point Theory, 3 (2013), No. 2, 406-417 ISSN: 1927-6303 [7] S. Hizarci. A fixed point theorem for triangular surface mapping. International Journal of Academic Research Part A; 2014; 6(3),178-180. DOI: 10.7813/2075-4124.2014/6-3/A.24 |
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AbstractIn this paper, we observe some new spaces to obtain new β- and γ- type duality of a sequence space λ, related to the some sequence spaces. Before this we give some new distinguished subspaces of an FK space obtained by an operator of Ayd n and Başar [2], which is stronger than common C1- Cesàro oper- ator. We also give some structural theorems and inclusions for these distinguished subspaces. Finally we prove some theorems related to the f-, ars- and arb- duality of a sequence space λ like Goes [14] and Buntinas [8]. These theorems are important to decade the duality of a sequence space in summability theory and topological sequence spaces theory.Keywords : FK spaces, Matrix methods, β-, γ-, f- duality. |
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References[1] B. Altay and F. Başar, Certain topological properties and duals of the matrix domain of a triangle matrix in a sequence space, J. Math. Anal. Appl. 336(1)(2007), 632 645. [2] C. Ayd n and F. Başar, On the new sequence spaces which include the spaces c0 and c, Hokkaido Math. J. 33(1)(2004), 1 16. [3] F. Başar, A note on the triangle limitation methods, F rat Üniv. Fen Müh. Bil. Dergisi, 5 (1) (1993), 113-117. [4] F. Başar, Summability Theory and Its Applications, Bentham Science Publishers, Istanbul, 2012. [5] J. Boos, Classical and Modern Methods in Summability, Oxford University Press. New York, Oxford, 2000. [6] M. Buntinas, Convergent and bounded Cesàro sections in FK-spaces, Math. Zeitschr., 121 (1971), 191-200. [7] M. Buntinas, On sectionally dense summability elds, Math. Zeitschr., 132 (1973), 141-149. [8] M. Buntinas, On Toeplitz sections in sequence spaces, Math. Proc. Camb. Phil. Soc., 78 (1975), 451-460. [9] . Dag adur, On Some subspaces of an FK space, Mathematical Communications, 7 (2002), 15-20. [10] R. Devos, Combinations of distinguished subsets and conullity, Math. Zeitschr., 192 (1986), 447-451. [11] D. J. H. Garling, The β- and γ-duality of sequence spaces, Proc. Camb. Phil. Soc., 63 (Jan. 1967), 963-981. [12] D. J. H. Garling, On topological sequence spaces, Proc. Camb. Phil. Soc., 63 (1967), 997-1019. 9 [13] G. Goes and S. Goes, Sequences of bounded variation and sequences of fourier coe cients. I, Math. Zeitschr.,118(1970), 93-102. [14] G. Goes, Summan von FK-räumen funktionale abschnittskonvergenz und umkehrsatz, Tohoku. Math. J., 26(1974), 487-504. [15] E. Malkowsky, Recent results in the theory of matrix transformations in sequence spaces, Mat. Vesnik 49(1997), 187 196. [16] A. Wilansky, Functional Analysis, Blaisdell Press, 1964. [17] A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw Hill, New York, 1978. [18] A. Wilansky, Summability Through Functional Analysis, North-Holland, Amsterdam, 1984. [19] K. Zeller, Allgemeine eigenschaften von limitierungsverfahren, Math. Zeitschr., 53 (1951), 463-487. |
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AbstractIn this study we try to show convergence of the Picard-Mann hybrid Iteration in convex cone metric spaces for common fixed points of infinite families of uniformly quasi-Lipschitzian mappings and quasi-nonexpansive mappings. A convex cone mtric space is a cone metric space with a convex structure.Keywords : convex metric space, convex structure, convex cone metric spaces, Picard-Mann hybrid iteration |
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References[1] W. Takahashi, A convexity in metric space and nonexpansive mappings, Kodai. Math. Sem.Rep. 22 (1970) 142-149. [2] Y. X. Tian, Convergence of an Ishikawa type iterative scheme for asymptotically quasi-nonexpansive mappings, Comput. Math. Appl. 49 (2005) 1905-1912. [3] C. Wang, L. W. Liu, Convergence theorems for fixed points of uniformly uniformly quasi-Lipschitzian mappings in convex metric spaces, Nonlinear Anal. TMA 70 (2009) 2067-2091. [4] S.S. Chang, L. Yang, X. R. Wang, Stronger convergence theorems for an infinite family of uniformly quasi-Lipschitzian mappings in convex metric spaces, Appl. Math. Comp. 217 (2010) 277-282. [5] Q. Y. Liu, Z. B. Liu, N. J. Huang, Approximating the common fixed points of two sequences of uniformly quasi-Lipschitzian mappings in convex metric spaces, Appl. Math. Comp. 216 (2010) 883-889. [6] B. S. Lee, Strong convergence theorems with a Noor-type iterative scheme in convex metric spaces, Com. Math. Appl. 61 (2011) 3218-3225. [7] Q. H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with errors number, J. Math. Anal. Appl. 259 (2001) 18-24. [8] S. H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Theory and Applications, Open acces, doi:10.1186/1687-1812-2013, 69. [9] L.-G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476. [10] S. Elmas and M. Ozdemir, Convergence of a general iterative scheme for three families of uniformly quasi-Lipschitzian mappings in convex metric spaces, Advances in Fixed Point Theory 3 (2013) 406-417. |
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AbstractAnalytical solution of highly nonlinear system of two dimensional Volterra integral equations is studied by the reduced dierential transform method [RDTM]. We present a new property of RDTM to acquire the recursive relation which is used to get analytical solution of the above mentioned two dimensional system. Results of the nu- merical examples obtained by RDTM are compared with the existing results obtained by TDDTM. Though solutions obtained by RDTM and TDDTM are same, RDTM has signicant advantage over TDDTM that is RDTM generates the solution of the nonlinear problem by operating the multivariable function with respect to a desired variable only not on all of their independent variables unlike in TDDTM so that RDTM reduces the time consumption than TDDTM.Keywords : Reduced dierential transform method, Volterra integral equa- tions |
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References[1] M.V.K. Chari and S.J. Salon, Numerical Methods in Electromagnetism, Academic Press (2000). [2] A. J. Jerri, Introduction to Integral Equations with Applications, 2nd ed.Wiley, New York (1999). [3] T. S. Sankar and V. I. Fabrikant, Investigations of a two-dimensional integral equation in the theory of elasticity and electrostatics, J. Mec. Theor. Appl., 2 (1983) 285{299. [4] V. M. Aleksandrov and A. V. Manzhirov, Two-dimensional integral equations in applied mechanics of deformable solids, J. Appl. Mech. Tech. Phys., 5 (1987) 146{152. [5] A. V. Manzhirov, Contact problems of the interaction between viscoelastic foundations subject to ageing and systems of stamps not applied simultaneously, Prikl. Matem. Mekhan., 4 (1987) 523{535. [6] Q. Tang and D. Waxman, An integral equation describing an asexual population in a changing environment, Nonlinear Anal., 53 (2003), 683{699. [7] A. Tari, On the existence uniqueness and solution of the nonlinear Volterra partial integro-dierential Equations, Inter. J. Nonlinear Sci., 16 (2013), no.2, 152{163. [8] B. G. Pachpatte, Volterra integral and integro dierential equations in two variables, J. Inequal. Pure Appl. Math., 10 (2009), no. 4, 1{10. [9] G. Q. Han and L. Q. Zhang, Asymptotic error expansion of two-dimensional Volterra integral equa- tion by iterated collocation, Appl. Math. Comput., 61 (1994), no. 2-3, 269{285. [10] M.Kwapisz, Weighted norms and existence and uniqueness of Lp solutions for integral equations in several variables, J. Dier. Equ., 97 (1992), 246-262. [11] R. Abazari and A. Klman, Numerical study of two-dimensional Volterra integral equations by RDTM and comparison with DTM, Abstr. Appl. Anal., 2013, Art. ID 929478, 10 pp. [12] H. Brunner and J.P. Kauthen, The numerical solution of two-dimensional Volterra integral equations by collocation and iterated collocation, IMA J. Numer. Anal., 9 (1989), 47{59. [13] M. Hadizadeh and N. Moatamedi, A new dierential transformation approach for two-dimensional Volterra integral equations, Inter. J. Comput. Math., 84 (2007), no. 4, 515{526. [14] A. Tari, M.Y. Rahimi, S. Shahmorad and F. Talati, Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the dierential transform method, J. Comput. Appl. Math., 228 (2009), no. 1, 70{76. [15] B. Jang, Comments on "Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the dierential transform method", J. Comput. Appl. Math., 233 (2009), no. 2, 224{ 230. [16] M. Tavassoli Kajani and N. Akbari Shehni, Solutions of two dimensional integral equation systems by dierential transform method, Appl. Math. Comput. Eng., ISBN: 978-960-474-270-7, 74{77. [17] Y. Keskin and G. Oturanc, Reduced dierential transform method for partial dierential equations, Int. J. Nonlinear Sci. Numer. Simul., 10 (2009), no. 6, 741{749. [18] Y. Keskin and G. Oturanc, Reduced dierential transform method for solving linear and nonlinear wave equations, Iran. J. Sci. Technol. Trans. A Sci., 34 (2010), no. 2, 113{122. AN APPLICATION OF RDTM 10 [19] A. Saravanan and N. Magesh, A comparison between the reduced dierential transform method and the Adomian decomposition method for the Newell-Whitehead-Segel equation, J. Egyptian Math. Soc., 21 (2013), no. 3, 259{265. [20] N. Magesh and A. Saravanan, The reduced dierential transform method for solving the systems of two dimensional nonlinear Volterra integrodierential equations, Proc. Int. Conf. Math. Sci., Elsevier, (2014), 217-220. [21] A. Saravanan and N. Magesh, An ecient computational technique for solving the FokkerPlanck equation with space and time fractional derivatives, J. King Saud Univ. Sci., 28 (2016) no. 2, 160{ 166. [22] O. Acan, M. M. Al Qurashi and D. Baleanu, New exact solution of generalized biological population model, J. Nonlinear Sci. Appl., 10 (2017), no. 7, 3916{3929. |
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AbstractIn the present study, one-dimensional advection–diffusion equation with constant coefficients is solved using Galerkin Method. We give the generlized solution of this equation. Two examples are presented for the numerical solution of this equation and results are compared with exact solution.Keywords : Univalent function, bi-univalent function, Coeffcient bounds. |
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References[1] Kumar A., Jaiswal D. K., Kumar N., Analytical solutions to one-dimensional advection–diffusion equation with variable coefficients in semi-infinite media , Journal of Hidrology, 330-337, 2010. [2] Srivastava, R., Flow Through Open Channels, Oxford University Press, 2008. [3] Bahar E., Gurarslan G., Numerical Solution of Advection-Diffusion Equation Using Operator Splitting Method, International Journal of Engineering & Applied Sciences , 9(4), 76-88, 2017. [4] Hasanov A., Simultaneous determination of the source terms in a linear hyperbolic problem from the final over determination: weak solution approach, IMA Journal of Applied Mathematics, 74, 1-19, 2008. [5] Ladyzhenskaya O.A., Boundary Value Problems in Mathematical Physics, Springer-Verlag, 1985. [6] Li, Q.H., Wang J., Weak Galerkin Finite Element methods for parabolic equations, Numerical Methods for Partial Differential Equations, 29,2004-2024, 2013. [7] Dedner, A., Madhavan P., Stinner B., Analysis of the discontinuous Galerkin method for elliptic problems on surfaces, IMA Journal of Numerical Analysis, 33, 952-973, 2013. [8] Huang Y., Li J., Li D., Developing weak Galerkin finite element methods for the wave equation, Numerical Methods for Partial Differential Equation, 33,3,2017. [9] Sengupta T.K., Talla S.B., Pradhan S.C., Galerkin finite element methods for wave problems, Sadhana, 30, 5, 611–623, 2005. |
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AbstractThe aim of this investigation is to give a new subclass of analytic functions dened by Salagean dierential operator and nd upper bound of Zalcman functional a2 nKeywords : Univalent function, bi-univalent function, Coecient bounds. |
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References[1] D. Bansal and J. Sokol, Zalcman conjecture for some subclass of analytic functions, J. Fract. Calc. Appl., Vol. 8(1) Jan. 2017, pp. 1-5. [2] J.E. Brown and A. Tsao, On the Zalcman conjecture for starlikeness and typically real functions, Math. Z., 191 (1986), 467474. [3] P.L. Duren, Univalent Functions, Grundlehren der Mathematischen Wis- senschaften, Vol. 259. Springer:New York, NY,USA, 1983. [4] A.E. Livingston, The coecients of multivalent close-to-convex functions, Proc. Amer. Math. Soc., 21 (1969), 545552. [5] W. Ma, The Zalcman conjecture for close-to-convex functions, Proc. Amer. Math. Soc., 104(1988), 741744. [6] J. Nishiwaki, S. Owa, Coecient inequalities for certain analytic functions, Int. J. Math. Math. Sci. 29(2002) 285290. [7] M. Nunokawa,A sucient condition for univalence and starlikeness, Proc. Japan Acad. Ser. A., 65(1989) 163164. [8] C. Pommerenke, Univalent Functions. Gottingen, Germany: Vandenhoeck and Rupercht, 1975. [9] H. Saitoh, M. Nunokawa, S. Fukui, S. Owa, A remark on close-to-convex and starlike functions, Bull. Soc. Roy. Sci. Liege, 57(1988) 137141. [10] G.S. Salagean, Subclasses of univalent functions, in Complex Analysis, Fifth Romanian-Finnish Seminar, Vol. 1013 of Lecture Notes in Mathematics, pp. 362- 372, Springer, Berlin, Germany, 1983. [11] R. Singh, S. Singh, Some sucient conditions for univalence and starlikeness Collect. Math., 47(1982) 309314. [12] B.A. Uralegaddi, M.D. Ganigi, S. M. Sarangi, Univalent functions with positive coecients Tamkang J. Math., 25(1994) 225230 |
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AbstractPurpose of this paper is to determine some regular non-extendible D(n) triples for some fixed integer n. Besides, paper includes a number of algebraic properties for such diophantine sets with size three.Keywords : Diophantine Sets, Property D(n), Integral Solutions of Pell Equations, Quadratic Residues nd Reciprocity, Legendre Symbol. |
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ReferencesBiggs N.L., Discrete Mathematics. Oxford University Press, 2003. Burton D.M., Elementary Number Theory. Tata McGraw-Hill Education, 2006. Clark, P. L., Quadratic Reciprocity I, Lecture Notes, 1-12. Cohen H., Number Theory, Graduate Texts in Mathematics, vol. 239, Springer-Verlag, New York, 2007. Daileda, R.C., The Legendre Symbol, Lecture Notes in Number Theory, Trinity University. Dickson LE., History of Theory of Numbers and Diophantine Analysis, Vol 2, Dove Publications, New York, 2005. Dotsenko, V., Quadratic Residues, Lecture Notes, pp. 1-3. Dujella, A., Jurasic, A., Some Diophantine Triples and Quadruples for Quadratic Polynomials, J. Comp. Number Theory, Vol.3, No.2, 123-141, 2011. Fermat, P. Observations sur Diophante, Oeuvres de Fermat, Vol.1 (P. Tonnery, C. Henry eds.), p.303, 1891. Gopalan M.A., Vidhyalaksfmi S. , Özer Ö., “A Collection of Pellian Equation (Solutions and Properties)”, Akinik Publications, New Delhi, INDIA, 2018. Ireland K. and Rosen M., A Classical Introduction to Modern Number Theory, 2nd ed., Graduate Texts in Mathematics,vol. 84, Springer-Verlag, New York, 1990. Jimbo, H.C. and Ngongo, I.S., Application of Limit Theorem to sum of Legendre Symbols, Acta Math. Univ. Comenianae, Vol. (submitted to AMUC), pp. 1-5. Kohnen, W., An Elemenary Proof in the Theory of Quadratic Resıdues, Bull. Korean Math. Soc. 45 (2008), No. 2, pp. 273-275. Lemmermeyer, F. Reciprocity Laws: from, Euler to Eisenstien. Berlin Hiedelberg New York: Springer-Verlag,2000. Mathmdmb, Primitive Roots,Order, Quadratic Residue, Lecture Notes, March 30, 2011. Mollin R.A., Fundamental Number theory with Applications, CRC Press, 2008. Özer Ö., A Note On The Particular Sets With Size Three, Boundary Field Problems and Computer Simulation Journal, 55, 56-59, 2016. Özer Ö., On The Some Particular Sets, Kırklareli University Journal of Engineering and Science, 2, 99-108, 2016. Özer Ö., Some Properties of The Certain Pt Sets, International Journal of Algebra and Statistics, Vol. 6; 1-2 , 117-130, 2017. Özer Ö., On The Some Nonextandable Regular P_(-2) Sets , Malaysian Journal of Mathematical Sciences 12(2): 255–266, 2018. Özer Ö., Dubey O.P., On Some Particular Regular Diophantine 3-Tuples, Mathematics in Natural Sciences, (Accepted). Stillwell, J. Elements of Number Theory. New York: Springer-Verlag, 2003. |
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AbstractIn this study, we have shown that there are different solutions of an angle question in the triangle of secondary educationKeywords : Angle Question ,Triangle an d Euclid Geometry |
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References[1] G. D. Birkhoff and R. Beatley, Basic Geometry, AMS Chelsea Publ., 2000, 3 rd edition [2] D. A. Brannan, M. F. Esplen, J. J. Gray, Geometry, Cambridge University Press, 2002 [3] J. N. Cederberg, A Course in Modern Geometries, Springer, 1989 [4] A. De Morgan, On the Study and Difficulties of Mathematics, Dover, 2005 [5] D. Hilbert, Foundations of Geometry, Open Court, 1999 [6] T. Hobbes, Leviathan, ch. 46, Penguin Classics, 1982 B. Jowett, The 3.2 Other solutions [7] Dialogues of Plato, Random House, 1982 [8] F. Klein, Elementary Mathematics from an Advanced Standpoint: Geometry, Dover, 2004 [9] D. Pedoe, Geometry: A Comprehensive Course, Dover, 1988 [10] C. Pritchard (ed.), The Changing Shape of Geometry, Cambridge University Press, 2003 [11] S. Roberts, King of Infinite Space, Walker & Company, 2006 [12] S.Hizarci, S.Elmas. A.Kaplan, A.S.Ipek, C.Isık, Düzlem Geometri,Palme Yayıncılık, Ankara,2009 [13] S .Elmas ,Golden Ratio in Plain and Spehere,International Journal Of Academic Research, 2015 [14] S.Elmas , S. Hizarci .A Different Look at the Pol Erdos-Mordell , International Journal of Science and Technology Volume 2 No. 4, April, 201 [15] S.Elmas , S. Hizarci , Altın Oranın Farklı Formu ,Journal of the Fine Arts Institute (GSED),Number 38, Erzurum, 2017, 99-10 16 [16] S.Elmas , S. Hizarci,The den Ratio, the Golden Number and a review on Fixed Point,Ciencia e Tecnica Vitivinicola. ISSN:0254-0223 Vol. 31 ,n. 12.2016. [17] Kaplan.A, Tortumlu .N,Hizarci. S, A Simple Construction of the Golden Ratio, World Applied Sciences Journal 7(7),833-833.2009. |
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