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2014
Collected papers, Vol. V
Florentin Smarandache
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Part of the Mathematics Commons, Non-linear Dynamics Commons, Other Applied Mathematics Commons, and the Special Functions Commons Collected Papers, V Florentin Smarandache
Papers of Mathematics or Applied mathematics
Brussels, 2014 Florentin Smarandache
Collected Papers
Vol. V
Papers of Mathematics or Applied mathematics
EuropaNova Brussels, 2014 PEER REVIEWERS: Prof. Octavian Cira, Aurel Vlaicu University of Arad, Arad, Romania Dr. Stefan Vladutescu, University of Craiova, Craiova, Romania Mumtaz Ali, Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan Siad Broumi, Faculty of Arts and Humanities, Hay El Baraka Ben M'sik Casablanca, Morocco
Cover: Conversion or transfiguration of a cylinder into a prism.
EuropaNova asbl 3E, clos du Parnasse 1000, Brussels Belgium www.europanova.be
ISBN 978-1-59973-317-3 TABLE OF CONTENTS
MATHEMATICS ...... 13
Florentin Smarandache, Cătălin Barbu The Hyperbolic Menelaus Theorem in The Poincaré Disc Model Of Hyperbolic Geometry 14
Florentin Smarandache, Cătălin Barbu A new proof of Menelaus’s Theorem of Hyperbolic Quadrilaterals in the Poincaré Model of Hyperbolic Geometry 20
Ion Pătraşcu, Florentin Smarandache Some Properties of the Harmonic Quadrilateral 26
Ion Pătraşcu, Florentin Smarandache Non-Congruent Triangles with Equal Perimeters and Arias 32
Florentin Smarandache Another proof of the I. Patrascu’s theorem 35
Octavian Cira, Florentin Smarandache Luhn Prime Numbers 37 Mircea E. Şelariu, Florentin Smarandache, Marian Niţu Cardinal Functions And Integral Functions 45
Mihály Bencze, Florentin Smarandache About an Identity and its Applications 58
Florentin Smarandache On Crittenden and Vanden Eynden’s Conjecture 60
Marian Niţu, Florentin Smarandache, Mircea E. Şelariu Eccentricity, Space Bending, Dimension 61
Florentin Smarandache Professor Şelariu’s Supermathematics 71
Marian Niţu, Florentin Smarandache, Mircea E. Şelariu Excentricitatea, dimensiunea de deformare a spaţiului 81
Mircea E. Şelariu, Florentin Smarandache, Marian Niţu Funcţii cardinale şi funcţii integrale circulare excentrice 103
Florentin Smarandache SuperMatematica Profesorului Şelariu
121 EXTENICS ...... 131
Xingsen Li, Yingjie Tian, Florentin Smarandache, Rajan Alex An Extension Collaborative Innovation Model in The Context of Big Data 132
Victor Vladareanu, Florentin Smarandache, Luige Vladareanu Extension Hybrid Force-Position Robot Control in Higher Dimensions 155
Florentin Smarandache, Ştefan Vlăduţescu Extension communication pentru rezolvarea contradicţiei ontologice dintre comunicare şi informaţie 165
Florentin Smarandache, Tudor Păroiu Extenica 177 MECHATRONICS...... 194
Luige Vlădăreanu, Gabriela Tonţ, Victor Vlădăreanu, Florentin Smarandache The navigation of mobile robots in non-stationary and non-structured environments 195
Luige Vlădăreanu, Gabriela Tonţ, Victor Vlădăreanu, Florentin Smarandache, Lucian Căpitanu The Navigation Mobile Robot Systems Using Bayesian Approach through the Virtual Projection Method 205
Kimihiro Okuyama, Mohd Anasri, Florentin Smarandache, Valeri Kroumov Mobile Robot Navigation Using Artificial Landmarks and GPS 211
STATISTICS...... 217
Mukesh Kumar, Rajesh Singh, Ashish K. Singh, Florentin Smarandache Some Ratio Type Estimators 218
Manoj K. Chaudhary, Rajesh Singh, Rakesh K. Shukla, Mukesh Kumar, Florentin Smarandache A Family of Estimators for Estimating Population Mean in Stratified Sampling under Non-Response 223 Rajesh Singh, Sachin Malik, A. A. Adewara, Florentin Smarandache Multivariate Ratio Estimation With Known Population Proportion of Two Auxiliary Characters For Finite Population 231
V.V. Singh, Alka Mittal, Neetish Sharma, Florentin Smarandache Determinants of Population Growth in Rajasthan: An Analysis 239
Rajesh Singh, Mukesh Kumar, Florentin Smarandache Ratio Estimators in Simple Random Sampling when Study Variable is an Attribute 251
Jayant Singh, Hansraj Yadav, Florentin Smarandache Rural Migration A Significant Cause Of Urbanization: A District Level Review Of Census Data For Rajasthan 255
Jayant Singh, Hansraj Yadav, Florentin Smarandache Urbanization due to Migration: A District Level Analysis of Migrants from Different Distances for The Rajasthan State 262 MISCELLANEA...... 272
Florentin Smarandache Administration, Teaching and Research Philosophies 273
Florentin Smarandache, Ştefan Vlăduţescu An Application of The Systemic Theory in The Field of Industrial Companies 281
V. Christianto, Florentin Smarandache On Gödel's incompleteness theorem(s), Artificial Intelligence/Life, and Human Mind 285
Priti Singh, Florentin Smarandache, Dipti Chauhan, Amit Bhaghel A Unit Based Crashing Pert Network for Optimization of Software Project Cost 293
Florentin Smarandache Çok Kriterli Karar Verme için Alfa İndirgeme Yöntemi (α-İ ÇKKV) 303
Florentin Smarandache Recreational Mathematics: Puzzle Me! 325 Florentin Smarandache A Scientist and Haiku Poet 327
Florentin Smarandache Review of the journal „Us and the Sky” 329
Dmitri Rabounski Florentin Smarandache: polymath, professor of mathematics 330 AUTHORS
Mathematics Cătălin Barbu, 14-19, 20-25 Mihály Bencze, 58-59 Octavian Cira, 37-44 Marian Niţu, 45-57, 61-70, 81-102, 103-120 Ion Pătraşcu, 26-31, 32-34 Florentin Smarandache, 14-19, 20-25, 26-31, 32-34, 35-36, 37-44, 45-57, 58-59, 60, 61-70, 71-80, 81-102, 103-120, 121-130 Mircea E. Şelariu, 45-57, 61-70, 81-102, 103-120
Extenics Rajan Alex, 132-154 Xingsen Li, 132-154 Tudor Păroiu, 177-193 Florentin Smarandache, 132-154, 155-164, 165-176, 177-193 Luige Vlădăreanu, 155-164 Victor Vlădăreanu, 155-164 Ştefan Vlăduţescu, 165-176 Yingjie Tian, 132-154
Mechatronics Mohd Anasri, 211-216 Lucian Căpitanu, 205-210 Valeri Kroumov, 211-216 Kimihiro Okuyama, 211-216 Florentin Smarandache, 195-204, 205-210, 211-216 Luige Vlădăreanu, 195-204, 205-210 Victor Vlădăreanu, 195-204, 205-210 Gabriela Tonţ, 195-204, 205-210 Statistics A. A. Adewara, 231-238 Manoj K. Chaudhary, 223-230 Mukesh Kumar, 218-222, 223-230, 251-254 Sachin Malik, 231-238 Alka Mittal, 239-250 Neetish Sharma, 239-250 Rakesh K. Shukla, 223-230 Ashish K. Singh, 218-222 Jayant Singh, 255-261, 262-271 Rajesh Singh, 218-222, 223-230, 231-238, 251-254 V.V. Singh, 239-250 Florentin Smarandache, 218-222, 223-230, 231-238, 239-250, 251-254, 255-261, 262-271 Hansraj Yadav, 255-261, 262-271
Miscellanea Amit Bhaghel, 293-302 Dipti Chauhan, 293-302 V. Christianto, 285-292 Dmitri Rabounski, 330-334 Priti Singh, 293-302 Florentin Smarandache, 273-280, 281-284, 285-292, 293-302, 303-324, 325-326, 327-328, 329 Ştefan Vlăduţescu, 281-284 Florentin Smarandache Collected Papers, V Introductory Note
This volum includes 37 papers of mathematics or applied mathematics written by the author alone or in collaboration with the following co-authors: Cătălin Barbu, Mihály Bencze, Octavian Cira, Marian Niţu, Ion Pătraşcu, Mircea E. Şelariu, Rajan Alex, Xingsen Li, Tudor Păroiu, Luige Vlădăreanu, Victor Vlădăreanu, Ştefan Vlăduţescu, Yingjie Tian, Mohd Anasri, Lucian Căpitanu, Valeri Kroumov, Kimihiro Okuyama, Gabriela Tonţ, A. A. Adewara, Manoj K. Chaudhary, Mukesh Kumar, Sachin Malik, Alka Mittal, Neetish Sharma, Rakesh K. Shukla, Ashish K. Singh, Jayant Singh, Rajesh Singh,V.V. Singh, Hansraj Yadav, Amit Bhaghel, Dipti Chauhan, V. Christianto, Priti Singh, and Dmitri Rabounski.
They were written during the years 2010-2014, about the hyperbolic Menelaus theorem in the Poincare disc of hyperbolic geometry, and the Menelaus theorem for quadrilaterals in hyperbolic geometry, about some properties of the harmonic quadrilateral related to triangle simedians and to Apollonius circles, about Luhn prime numbers, and also about the correspondences of the eccentric mathematics of cardinal and integral functions and centric mathematics, or ordinary mathematics; there are some notes on Crittenden and Vanden Eynden's conjecture, or on new transformations, previously non-existent in traditional mathematics, that we call centric mathematics (CM), but that became possible due to the new born eccentric mathematics, and, implicitly, to the supermathematics (SM); also, about extenics, in general, and extension innovation model and knowledge management, in particular, about advanced methods for solving contradictory problems of hybrid position-force control of the movement of walking robots by applying a 2D Extension Set, or about the notion of point-set position indicator and that of point-two sets position indicator, and the navigation of mobile robots in non-stationary and nonstructured environments; about applications in statistics, such as estimators based on geometric and harmonic mean for estimating population mean using information; about Godel’s incompleteness theorem(s) and plausible implications to artificial intelligence/life and human mind, and many more.
References:
Florentin Smarandache: Collected Papers, Vol. I (first edition 1996, second edition 2007) http://fs.gallup.unm.edu/CP1.pdf Florentin Smarandache: Collected Papers, Vol. II (Chişinău, Moldova, 1997) http://fs.gallup.unm.edu/CP2.pdf Florentin Smarandache: Collected Papers, Vol. III (Oradea, Romania, 2000) http://fs.gallup.unm.edu/CP3.pdf Florentin Smarandache: Collected Papers, Vol. IV (100 Collected Papers of Sciences). Multispace & Multistructure. Neutrosophic Transdisciplinarity (Hanko, Finland, 2010) http://fs.gallup.unm.edu/MultispaceMultistructure.pdf Florentin Smarandache Collected Papers, V
MATHEMATICS
13 Florentin Smarandache Collected Papers, V
THE HYPERBOLIC MENELAUS THEOREM IN THE POINCARE´ DISC MODEL OF HYPERBOLIC GEOMETRY
FLORENTIN SMARANDACHE, CA�TA�LIN BARBU
Abstract. In this note, we present the hyperbolic Menelaus theorem in the Poincar´e disc of hyperbolic geometry. Keywords and phrases: hyperbolic geometry, hyperbolic triangle, gyrovector. 2000 Mathematics Subject Classification: 30F45, 20N99, 51B10, 51M10.
1. Introduction
Hyperbolic Geometry appeared in the first half of the 19th century as an attempt to understand Euclid’s axiomatic basis of Geometry. It is also known as a type of non-Euclidean Geometry, being in many respects similar to Euclidean Geometry. Hyperbolic Geometry includes similar concepts as distance and angle. Both these geometries have many results in common but many are different. There are known many models for Hyperbolic Geometry, such as: Poincar´e disc model, Poincar´ehalf-plane, Klein model, Einstein relativistic velocity model, etc. The hyperbolic geometry is a non-euclidian geometry. Menelaus of Alexandria was a Greek mathematician and astronomer, the first to recognize geodesics on a curved surface as natural analogs of straight lines. Here, in this study, we present a proof of Menelaus’s theorem in the Poincar´edisc model of hyperbolic geometry.
14 Florentin Smarandache Collected Papers, V
The well-known Menelaus theorem states that if l is a line not through any vertex of a triangle ABC such that l meets BC in D,CA in E, and AB in F , then DB EC FA · · = 1 [1]. This result has a simple statement but it is of great DC EA F B interest. We just mention here few different proofs given by A. Johnson [2], N.A. Court [3], C. Co¸snit¸˘a[4], A. Ungar [5]. F. Smarandache (1983) has generalized the Theorem of Menelaus for any polygon with n ≥ 4 sides as follows: If a line l intersects the n-gon A1A2...An sides A1A2,A2A3, ..., and AnA1 respectively in the M1A1 M2A2 MnAn points M1,M2, ..., and Mn, then · · ... · = 1 [6]. M1A2 M2A3 MnA1 We begin with the recall of some basic geometric notions and properties in the Poincar´edisc. Let D denote the unit disc in the complex z-plane, i.e.
D = {z ∈ C : |z| < 1}.
The most general M¨obiustransformation of D is
iθ z0 + z iθ z → e = e (z0 ⊕ z), 1 + z0z which induces the M¨obiusaddition ⊕ in D, allowing the M¨obiustransformation of the disc to be viewed as a M¨obiusleft gyro-translation
z0 + z z → z0 ⊕ z = 1 + z0z
followed by a rotation. Here θ ∈ R is a real number, z, z0 ∈ D, and z0 is the complex conjugate of z0. Let Aut(D, ⊕) be the automorphism group of the grupoid (D, ⊕). If we define
a ⊕ b 1 + ab gyr : D × D → Aut(D, ⊕), gyr[a, b] = = , b ⊕ a 1 + ab then is true gyro-commutative law
a ⊕ b = gyr[a, b](b ⊕ a).
A gyro-vector space (G, ⊕, ⊗) is a gyro-commutative gyro-group (G, ⊕) that obeys the following axioms:
(1) gyr[u, v]a· gyr[u, v]b = a · b for all points a, b, u, v ∈G.
(2) G admits a scalar multiplication, ⊗, possessing the following properties. For all real numbers r, r1, r2 ∈ R and all points a ∈ G:
(G1) 1 ⊗ a = a
(G2) (r1 + r2) ⊗ a = r1 ⊗ a ⊕ r2 ⊗ a
(G3) (r1r2) ⊗ a = r1 ⊗ (r2 ⊗ a)
15 Florentin Smarandache Collected Papers, V
|r| ⊗ a a (G4) = kr ⊗ ak kak (G5) gyr[u, v](r ⊗ a) = r ⊗ gyr[u, v]a
(G6) gyr[r1 ⊗ v, r1 ⊗ v] =1
(3) Real vector space structure (kGk , ⊕, ⊗) for the set kGk of one-dimensional ”vectors” kGk = {± kak : a ∈ G} ⊂ R with vector addition ⊕ and scalar multiplication ⊗, such that for all r ∈ R and a, b ∈ G,
(G7) kr ⊗ ak = |r| ⊗ kak (G8) ka ⊕ bk ≤ kak ⊕ kbk.
Theorem 1 (The law of gyrosines in M¨obiusgyrovector spaces). Let ABC be a gyrotriangle in a M¨obiusgyrovector space (Vs, ⊕, ⊗) with vertices A, B, C ∈ Vs, sides a, b, c ∈ Vs, and side gyrolengths a, b, c ∈ (−s, s), a = ªB⊕C, b = ªC ⊕ A, c = ªA ⊕ B, a = kak , b = kbk , c = kck , and with gyroangles α, β, and γ at the vertices A, B, and C. Then a b c γ = γ = γ , sin α sin β sin γ v where v = [7, p. 267]. γ v2 1 − s2 Definition 2 The hyperbolic distance function in D is defined by the equation ¯ ¯ ¯ a − b ¯ d(a, b) = |a ª b| = ¯ ¯ . ¯1 − ab¯
Here, a ª b = a ⊕ (−b), for a, b ∈ D. For further details we refer to the recent book of A.Ungar [5].
2. Main results
In this section, we prove the Menelaus’s theorem in the Poincar´edisc model of hyperbolic geometry.
Theorem 3 (The Menelaus’s Theorem for Hyperbolic Gyrotriangle). If l is a gyroline not through any vertex of an gyrotriangle ABC such that l meets BC in D, CA in E, and AB in F, then (AF ) (BD) (CE) γ · γ · γ = 1. (BF )γ (CD)γ (AE)γ
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Proof. In function of the position of the gyroline l intersect internally a side of ABC triangle and the other two externally (See Figure 1), or the line l intersect all three sides externally (See Figure 2). If we consider the first case, the law of gyrosines (See Theorem 1), gives for the gyrotriangles AEF, BF D, and CDE, respectively (AE) sin AF[E (1) γ = , (AF )γ sin AE[F
(BF ) sin FDB\ (2) γ = , (BD)γ sin D\FB and (CD) sin DEC\ (3) γ = , (CE)γ sin E\DC where sin AF[ E = sin DF\B, sin EDC\ = sin FDB,\ and sin AEF[ = sin DEC,\ since gyroangles AEF[ and DEC\ are suplementary. Hence, by (1), (2) and (3), we have (AE) (BF ) (CD) sin AF[ E sin FDB\ sin DEC\ (4) γ · γ · γ = · · = 1, (AF )γ (BD)γ (CE)γ sin AE[F sin D\FB sin E\DC the conclusion follows. The second case is treated similar to the first.
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Naturally, one may wonder whether the converse of the Menelaus theorem exists.
Theorem 4 (Converse of Menelaus’s Theorem for Hyperbolic Gyro- triangle). If D lies on the gyroline BC,E on CA, and F on AB such that (AF ) (BD) (CE) (5) γ · γ · γ = 1, (BF )γ (CD)γ (AE)γ then D,E, and F are collinear. Proof. Relabelling if necessary, we may assume that the gyropoint D lies beyond B on BC. If E lies between C and A, then the gyroline ED cuts the gyroside AB, at F 0 say. Applying Menelaus’s theorem to the gyrotriangle ABC and the gyroline E − F 0 − D, we get
0 (AF )γ (BD)γ (CE)γ (6) 0 · · = 1. (BF )γ (CD)γ (AE)γ
0 (AF )γ (AF )γ 0 From (5) and (6), we get = 0 . This equation holds for F = F . (BF )γ (BF )γ Indeed, if we take x := |ªA ⊕ F 0| and c := |ªA ⊕ B| , then we get c ª x = |ªF 0 ⊕ B| . For x ∈ (−1, 1) define x c ª x (7) f(x) = : . 1 − x2 1 − (c ª x)2 c − x x(1 − c2) Because cªx = , then f(x) = . Since the following equality 1 − cx (c − x)(1 − cx) holds c(1 − c2)(1 − xy) (8) f(x) − f(y) = (x − y), (c − x)(1 − cx)(c − y)(1 − cy) we get f(x) is an injective function and this implies F = F 0, so D,E,F are collinear. There are still two possible cases. The first is if we suppose that the gyropoint F lies on the gyroside AB, then the gyrolines DF cuts the gyrosegment AC in the gyropoint E0. The second possibility is that E is not on the gyroside AC, E lies beyond C. Then DE cuts the gyroline AB in the gyropoint F 0. In each case a similar application of Menelaus gives the result.
References [1] Honsberger, R., Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Washington, DC: Math. Assoc. Amer., 1995, 147. [2] Johnson, R.A., Advanced Euclidean Geometry, New York, Dover Publica- tions, Inc., 1962, 147.
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[3] Court, N.A., A Second Course in Plane Geometry for Colleges, New York, Johnson Publishing Company, 1925, 122.
[4] Cos¸nit¸a,˘ C., Coordonn´eesBarycentriques, Paris, Librairie Vuibert, 1941, 7.
[5] Ungar, A.A., Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity, Hackensack, NJ:World Scientific Publishing Co.Pte. Ltd., 2008, 565.
[6] Smarandache, F., G´en´eralisation du Th´eorˇcmede M´en´elaus, Rabat, Se- minar for the selection and preparation of the Moroccan students for the International Olympiad of Mathematics in Paris - France, 1983.
[7] Ungar, A.A., Analytic Hyperbolic Geometry Mathematical Foundations and Applications, Hackensack, NJ:World Scientific Publishing Co.Pte. Ltd., 2005.
[8] Goodman, S., Compass and straightedge in the Poincar´edisk, American Mathematical Monthly 108 (2001), 38–49.
[9] Coolidge, J., The Elements of Non-Euclidean Geometry, Oxford, Claren- don Press, 1909.
[10] Stahl, S., The Poincar´ehalf plane a gateway to modern geometry, Jones and Barlett Publishers, Boston, 1993.
[11] Barbu, C., Menelaus’s Theorem for Hyperbolic Quadrilaterals in The Ein- stein Relativistic Velocity Model of Hyperbolic Geometry, Scientia Magna, Vol. 6, No. 1, 2010, p. 19.
Published in „Italian Journal of Pure and Applied Mathematics”, No. 30, 2013, 6 p.
19 Florentin Smarandache Collected Papers, V
A NEW PROOF OF MENELAUS’S THEOREM OF HYPERBOLIC QUADRILATERALS IN THE POINCARÉ MODEL OF HYPERBOLIC GEOMETRY
CATALIN BARBU and FLORENTIN SMARANDACHE Abstract. In this study, we present a proof of the Menelaus theorem for quadrilaterals in hyperbolic geometry, and a proof for the transversal theorem for triangles
2000 Mathematics Subject Classi…cation: 51K05, 51M10
Key words: hyperbolic geometry, hyperbolic quadrilateral, Menelaus theorem, the transversal theorem, gyrovector
1. Introduction
Hyperbolic geometry appeared in the …rst half of the 19th century as an attempt to un- derstand Euclid’saxiomatic basis of geometry. It is also known as a type of non-euclidean geometry, being in many respects similar to euclidean geometry. Hyperbolic geometry includes similar concepts as distance and angle. Both these geometries have many results in common but many are di¤erent. Several useful models of hyperbolic geometry are studied in the literature as, for instance, the Poincaré disc and ball models, the Poincaré half-plane model, and the Beltrami-Klein disc and ball models [3] etc. Following [6] and [7] and earlier discoveries, the Beltrami-Klein model is also known as the Einstein relativistic velocity model. Menelaus of Alexandria was a Greek mathematician and astronomer, the …rst to recognize geodesics on a curved surface as natural analogs of straight lines. The well-known Menelaus theorem states that if l is a line not through any vertex of a triangle DB EC FA ABC such that l meets BC in D;CA in E, and AB in F , then DC EA FB = 1 [2]. Here, in this study, we give hyperbolic version of Menelaus theorem for quadrilaterals in the Poincaré disc model. Also, we will give a reciprocal hyperbolic version of this theorem. In [1] has been given proof of this theorem, but to use Klein’smodel of hyperbolic geometry. We begin with the recall of some basic geometric notions and properties in the Poincaré disc. Let D denote the unit disc in the complex z - plane, i.e.
D = z C : z < 1 : f 2 j j g The most general Möbius transformation of D is
i z0 + z i z e = e (z0 z); ! 1 + z0z which induces the Möbius addition in D, allowing the Möbius transformation of the disc to be viewed as a Möbius left gyro-translation
z0 + z z z0 z = ! 1 + z0z
20 Florentin Smarandache Collected Papers, V
followed by a rotation. Here R is a real number, z; z0 D; and z0 is the complex 2 2 conjugate of z0: Let Aut(D; ) be the automorphism group of the grupoid (D; ). If we de…ne a b 1 + ab gyr : D D Aut(D; ); gyr[a; b] = = ; ! b a 1 + ab then is true gyro-commutative law
a b = gyr[a; b](b a): A gyro-vector space (G; ; ) is a gyro-commutative gyro-group (G; ) that obeys the following axioms: (1) gyr[u; v]a gyr[u; v]b = a b for all points a; b; u; v G: (2) G admits a scalar multiplication, , possessing the following2 properties. For all
real numbers r; r1; r2 R and all points a G: (G1) 1 a = a 2 2
(G2) (r1 + r2) a = r1 a r2 a (G3) (r1r2) a = r1 (r2 a) r a a (G4) jrj a = a (G5) gyrk [uk; v](kr k a) = r gyr[u; v]a
(G6) gyr[r1 v; r1 v] =1 (3) Real vector space structure ( G ; ; ) for the set G of one-dimensional "vec- tors" k k k k G = a : a G R k k f k k 2 g with vector addition and scalar multiplication ; such that for all r R and a; b G; (G7) r a = r a 2 2 (G8) ka bk j aj k kb k k k k k k De…nition 1. The hyperbolic distance function in D is de…ned by the equation
a b d(a; b) = a b = : j j 1 ab