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Hyperbolic Valued Signed Measure International Journal of Mathematics Trends and Technology (IJMTT) – Volume 55 Number 7 - March 2018 Hyperbolic Valued Signed Measure Chinmay Ghosh¹, Sanjoy Biswas² , Taha Yasin³ ¹Kazi Nazrul University, Nazrul Road, Asansol-713340, Paschim Bardhaman, West Bengal, India. 2,3Guru Nanak Institute of Technology, 157/F, Nilgunj Road, Panihati, Sodepur, Kolkata-700114, West Bengal, India. Abstract: In this article we introduced two symbols and as extended hyperbolic numbers and modified D D the definition of hyperbolic valued measure. Also we defined hyperbolic valued signed measure and proved some theorems on it. AMS Subject Classification (2010) : 28A12. Keywords: Hyperbolic numbers, hyperbolic conjugation, idempotent representation, hyperbolic modulus, D- bounded, D-Cauchy sequence, D-valued measure, D-measure space, D-valued signed measure. I. INTRODUCTION Albert Einstein gave the theory of special relativity at the beginning of the 20th Century which is based on Lorentzian geometry. The hyperbolic numbers (or duplex numbers or perplex numbers), put Lorentzian geometry to an analogous mathematical concept with Euclidean geometry. The hyperbolic numbers play a similar role in the Lorentzian plane that the complex numbers do in the Euclidean plane. Different structures of these numbers can understood well in [2],[3].[5],[6],[7],and [9]. D. Alpay et. al. [1] gave an idea of hyperbolic valued probabilities. This idea motivate to construct hyperbolic valued measure. Before the present work R. Kumar et. al. [4] had given the similar concept which is modified here by introducing two symbols and as extended hyperbolic numbers. Also the concept of hyperbolic valued signed measure is introduced and the hyperbolic version of Hahn and Jordan decomposition theorems are proved in this article. The proofs in this article are based on the book of I. K. Rana [8]. II. BASIC DEFINATIONS The set of hyperbolic numbers D is defined as D = x yk : x , y R . where k is an imaginary element such that k²=1 but k∉R. D forms a commutative ring with respect to D and . D defined by ()()()()xyk xyk xx yyk 1 1D 2 2 1 2 1 2 ().()()()x yk x yk xx yy xy xyk 11D 22 1212 1221 A. Hyperbolic conjugation The conjugate of a hyperbolic numbers x y k is x yk . Different algebraic operations (additive, involutive and multiplicative) on D with respect to the hyperbolic conjugation are as follows: 1. 2. 3. () Note that, x22 y R The modulus of a hyperbolic number x y k D , is defined by ISSN: 2231-5373 http://www.ijmttjournal.org Page 515 International Journal of Mathematics Trends and Technology (IJMTT) – Volume 55 Number 7 - March 2018 22 x y R Any hyperbolic number with 0 is said to be invertible (or non-singular), and its inverse is given by 1 If 0 but xy22 0 then ( 0 ) is a zero divisor. The only zero divisors in D are those ( 0 ) for which 22 . We denote the set of zero divisors by . Thus, xy 0 D OD ={ =x+yk: 0, =x²-y²=0} B. Idempotent representation There are two important zero divisors in D, 11 and its conjugate 11. k k 22 22 Set 1 k and 1 k e 1 e 2 2 2 One can check that and ee121 ee12 0 2 2 and ee11 ee22 So these two elements are called mutually orthogonal idempotent elements in D. Thus {e₁,e₂} forms a basis of D The two sets D e. D and D e. D e1 1 e 2 2 are (principal) ideals in the ring D and they have the properties: DD 0 and ee12 DDDee 12 (1) Formula (1) is called the idempotent decomposition of D. Every hyperbolic number x y k can be written as =(x+y)e +(x-y)e =ve +v e ,v ,v R 1 2 1 1 2 2 1 2 (2) Formula (2) is called the idempotent representation of a hyperbolic number. It has a remarkable feature: the algebraic operations of addition, multiplication, taking of inverse, etc. can be realized component-wise. Observe that the sets D and D can be written as e1 e 2 D ={se:s R}=Re; D ={te :t R}=Re Remark1. One should keep in mind the following ee121 1 2 2 properties: a) D if and only if e1 e1 b) if and only if D e e 2 2 C. Partial order on D The two idempotent orthogonal axes divide the whole hyperbolic plane into four quadrants, namely D ={ve+ve:v,v1 1 2 2 1 2 0} D ={ve+ve:v1 1 2 2 1 0,v 2 0} D ={ve+ve:v,1 1 2 2 1v 2 0,} D ={ve+ve:v1 1 2 2 1 0,v 2 0} Observe that D D = 0 + A hyperbolic number ζ is said to be (strictly) positive if zÎ D \ {0} and (strictly) negative if zÎ D - \ {0} The set of non-negative hyperbolic numbers is also defined as D ={x+yk:x²-y² 0,x 0} ISSN: 2231-5373 http://www.ijmttjournal.org Page 516 International Journal of Mathematics Trends and Technology (IJMTT) – Volume 55 Number 7 - March 2018 On the realization of D⁺, M.E. Luna-Elizarraras et.al. [6] defined a partial order relation on D. For two hyperbolic numbers ζ₁,ζ₂ the relation is defined as if and only if D One can check that this D 12D 21 . relation is reflexive, transitive and anti-symmetric. Therefore D is a partial order relation on D. This partial order relation on D is an extension of the total order relation ≤ on R. We say ζ₁ ζ₂ if ζ₁ ζ₂ but D D D . Also we say if and if . If neither nor 12 21 D 12 D 21 D 12 12 D then and are not comparable. 21 D 1 2 Definition 1: For any hyperbolic number the D-valued (or hyperbolic valued) modulus of ζ is = v1 e 1 + v 2 e 2 defined by | | =|ve+ve| =|v|e+|v|e D DD1 1 2 2 1 1 2 2 where and are the usual modulus of real numbers. v1 v 2 Definition 2: A subset A of D is said to be D-bounded if ∃ M∈D⁺ such that M for any A D D Set A={x1 R: y R, xe+ye 1 2 A} . A2 ={y R: x R, xe+ye 1 2 A} It is immediate task to verify that if A is D-bounded then A₁ and A₂ are bounded subset of R. Definition 3: For a D-bounded subset A of D, the supremum of A with respect to the D-valued (or hyperbolic valued) modulus is defined by sup A=supA e +supA e D 1 1 2 2 Definition 4: A sequence of hyperbolic numbers is said to be convergent to ζ ∈ D if for ε ∈ D⁺\{0} ∃ k ∈ n n 1 N such that for all nk³ nDD Then we write, lim n n Definition 5: A sequence of hyperbolic numbers is said to be D-Cauchy sequence ζ ∈ D if for ε ∈ D⁺\{0} n n 1 ∃ N ∈N such that N m ND D for all m=1,2,3,..... Note that a sequence of hyperbolic numbers is said to be convergent if and only if it is a D-Cauchy n n 1 sequence. Definition 6: A hyperbolic series is convergent if and only if its partial sums is a D-Cauchy sequence, i.e., n n 1 for any ε ∈ D⁺\{0} ∃ N ∈N such that m for any m ∈ N. N k D k 1 D Definition 7: A hyperbolic series is D-absolutely convergent if the series || is convergent. n nD n 1 n 1 Every D-absolutely convergent series is convergent. III. Main result A. D-valued (or hyperbolic valued) measures: We introduce two symbols D and D as +D = ve+1 1 v 2 e 2 :v 1 = v 2 = + D = v1 e 1 + v 2 e 2 :v 1 = v 2 = - One can check that x y k D only if x = ±∞ and y ∈ R. If we take y = 0 then ζ = x= D if and only if x = ±∞. Thus D are the extension of ±∞ on the extended real number system. ISSN: 2231-5373 http://www.ijmttjournal.org Page 517 International Journal of Mathematics Trends and Technology (IJMTT) – Volume 55 Number 7 - March 2018 We call the two symbols D and D as plus hyperbolic infinity and minus hyperbolic infinity. The following two sets are of special interests: DD D DD D The extension of the algebraic operations and the partial order on D⁺∪D⁻ to DD 1. For every ζ ∈D⁺∪D⁻, DDDD 2. For every ζ ∈D⁺∪D⁻, () DD and () DD ()() DDD and ()() DDD 3. For every ζ ∈D⁺\{0}, xx()() DDD xx()() DDD 4. For every ζ∈D⁻\{0} xx()() DDD xx()() DDD Further, ( DD )0 ( )0 0 ()()() DDD Note that the relations DD () and DD () are not defined. If but we say and if but we say D D 0D DD D \0 D 0 . Also if D but we say and if DDD D DD D 0 D \0but D we say 0 DDD . Definition 8: Let X be a non empty set and S be the sigma algebra of subsets of X. A set function D : SD is said to be a D-valued (or hyperbolic valued) measures on S if (i) D ( ) 0 (ii) D is countably additive on S i.e., AA () D n D n n 1 n 1 Whenever A is a sequence of pairwise disjoint sets in S.
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