پديد آورنده :
رستم زاده، محفوظ
عنوان :
مدل بلترامي- كلاين هندسه ي هذلولوي با كاربردهاي آن در نظريه ي نسبيت خاص اينشتين
محل تحصيل :
اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي
صفحه شمار :
[نه]،112ص.: مصور
يادداشت :
ص.ع.به فارسي و انگليسي
استاد راهنما :
قهرمان طاهريان
استاد مشاور :
اعظم اعتماد دهكردي
توصيفگر ها :
صفحه ي مطلق , نظريه نسبيت خاص ; K , K- لوپ , جايروگروه , مثلثات , مساحت
تاريخ نمايه سازي :
25/4/93
استاد داور :
سيلويا پيانتا، بهروز ميرزا،مليحه يوسف زاده، محمدرضا رئوفي
كد ايرانداك :
ID643 دكتري
چكيده انگليسي :
Beltrami Klein Model of Hyperbolic Geometry with Applications to Einstein s Theory of Special Relativity Mahfouz Rostamzadeh m rostamzadeh@math iut ac ir June 2014 Doctor of Science in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Sayed Ghahreman Taherian taherian@cc iut ac ir Advisor Dr Azam Etemad Dehkordy ae110mat@cc iut ac ir 2010 MSC 51A25 51F05 51F20 20N05 Key words Hyperbolic Geometry Beltrami Klein Model Absolute Plane Special Relativity Theory K loop Gyrogroup Trigonometry Area Abstract In this dissertation the notions of K loops and gyrogroups and then the gyrovector space approach of A A Ungar are introduced Gyrocommutative gyrogroup also is known as K loop addition of gyrovectors and a scalar multiplication form a gyrovector space just as vector addition of vectors and a scalar multiplication form a vector space Gyrovector spaces provide the algebraic settings for hyperbolic geometry just as vector spaces provide the algebraic settings for Euclidean geometry General i e including non continuous and non Archimedean absolute planes have been classi ed in di erent ways e g by using Lambert Saccheri quadrangles e g 16 34 or coordinate systems cf 8 27 Here we consider the notion of quasi end a pencil determined by two lines which neither intersect nor have a common perpendicular an ideal point of 8 The cardinality of the quasi ends which are incident with a line is the same for all lines hence it is an invariant A of the plane A and can be used to classify absolute planes We prove that for hyperbolic planes consequently for the Beltrami Klein model ends and quasi ends are the same so A 2 cf 19 33 A A Ungar introduced gyrogeometry in full analogy with Euclidean geometry based on relativity addition of A Einstein with his gyrovector space approach cf 51 52 He de nes area based on defect but by his de nition the additive property of area is failed Another approach for defect is from H Karzel in 14 18 for the relation between the K loop and the defect of an absolute plane in the sense of 20 In this dissertation we are going to develop a formulary for the Beltrami Klein model of hyperbolic plane inside the unit disc D of the complex numbers C with geometric approach of Karzel cf 39 We introduce a systematical exact de nition for defect and area in the Beltrami Klein model of hyperbolic geometry Combining the ideas and methods of Karzel and Ungar give an elegant concept for defect and area in this model By our de nition the additive property of area is valid cf 35 In particular we give a rigorous and elementary proof for the defect formula stated in 51 Theorem 15 Also we use the isomorphism between the ordered elds R and 1 1 in BeltramiKlein model of hyperbolic geometry for similar results of Ungar in full analogous with Euclidean Geometry cf 36 43 At the end we presnet some applications of hyperbolic geometry in the Einstein s theory of special relativity 1
استاد راهنما :
قهرمان طاهريان
استاد مشاور :
اعظم اعتماد دهكردي
استاد داور :
سيلويا پيانتا، بهروز ميرزا،مليحه يوسف زاده، محمدرضا رئوفي
