Intuitionistic Fuzzy Ideals of Near Rings
|
|
- Ευδώρα Παχής
- 5 χρόνια πριν
- Προβολές:
Transcript
1 International Mathematical Forum, Vol. 7, 202, no. 6, Intuitionistic Fuzzy Ideals of Near Rings P. K. Sharma P.G. Department of Mathematics D.A.V. College Jalandhar city, Punjab, India Abstract In this paper, an attempt has been made to study some algebraic nature of intuitionistic fuzzy ideals of near ring N and their properties with the help of their (α, β) cut sets. Mathematics Subject Classification: 03F55, 03E72, 6Y30 Keywords: Intuitionistic fuzzy set (IFS), Intuitionistic fuzzy subgroup (IFSG), Near ring, Intuitionistic fuzzy ideal (IFI), (α,β) cut set, (Near ring)- Homomorphism. Introduction The idea of Intuitionistic fuzzy sets was introduced by Atanassov as a generalization of the notion of fuzzy sets. Biswas applied the concept of Intuitionistic fuzzy sets to the theory of groups and studied intuitionistic fuzzy subgroups of a group. The notion of an intuitionistic fuzzy R-subgroups of a near ring is given by Jun, Yon and Kim. Zhan Jianming and Ma Xueling also discussed the various properties on intuitionistic fuzzy ideals of near rings. Sharma studied intuitionistic fuzzy subgroups of a group with the help of their.(α,β)-cut sets. Here in this paper we study the properties of Intuitionistic fuzzy ideals of near ring with the help of their.(α,β)-cut sets. By a near ring, we mean a non-empty set N with two binary operations + and. satisfying the following axioms (i) ( N, + ) is a group (ii) ( N,. ) is a semigroup (iii) ( x + y ).z = x.z + y.z for all x, y, z N
2 770 P. K. Sharma Precisely speaking it is a right near ring because it satisfies the right distributive law. If the condition (iii) is replace by z.( x + y ). = z.x + z.y for all x, y, z N, then it is called left near ring. In this paper we use the word near ring instead of right near ring.we denote xy instead of x.y. A near ring N is called zero symmetric if x.0 = 0 for all x N. An ideal of a near ring N is a subset I of N such that (i) ( I, + ) is normal subgroup of ( N, + ) (ii) I N I (iii) y ( x + i ) yx I ; for all x, y N and i I Note that I is right ideal of N if I satisfies (i) and (ii), and I is left ideal of N if I satisfies (i) and (iii).we note that intersection of a family of right (resp. left) ideals in N is a right (resp. left) ideal in N. If N and N be two near rings. A map f : N N is called a (near ring) homomorphism if f (x + y) = f (x) + f (y) and f (xy) = f(x)f(y) for any x, y N.The homomorphic image of right ( resp. left ) ideal in N is right ( resp. left ) ideal. The detail of results about near rings can be found in the book Near rings by G. Pilz. An Intuitionistic fuzzy set A = {< x, μ A (x), ν A (x) > : x X }in X can be identified with an ordered pair ( μ A, ν A ) in I X I X. For sake of simplicity, we shall use the symbol A = ( μ A, ν A ) for the IFS A = {< x, μ A (x), ν A (x) > : x X }. Clearly, every fuzzy set μ A is an IFS of the form (μ A, μ A c ), where μ A c = - μ A. 2. Preliminaries Definition (2.)[4] An IFS A = ( μ A, ν A ) of a near ring N is called an Intuitionistic fuzzy subnear-ring of N if for all x, y N (i) μ A (x-y) min{μ A (x), μ A (y)} and ν A (x-y) max{ν A (x), ν A (y)} (ii) μ A (xy) min{μ A (x), μ A (y)} and ν A (xy) max{ν A (x), ν A (y)} Definition (2.2)[4] An IFS A = ( μ A, ν A ) of a near ring N is called an Intuitionistic fuzzy N-subgroup of N if for all x, y, n N (i) μ A (x-y) min{μ A (x), μ A (y)} and ν A (x-y) max{ν A (x), ν A (y)} (ii) μ A (nx) μ A (x) and ν A (nx) ν A (x) (iii) μ A (xn) μ A (x) and ν A (xn) ν A (x) If the condition (i) and (ii) holds, then A is called Intuitionistic fuzzy left N-subgroup of N and if the condition (i) and (iii) holds, then A is called Intuitionistic fuzzy right N-subgroup of N.
3 Intuitionistic fuzzy ideals of near rings 77 Definition (2.3)[3] An IFS A = ( μ A, ν A ) of a near ring N is called an Intuitionistic fuzzy Ideal of N if for all x, y, n N (i) μ A (x-y) min{μ A (x), μ A (y)} and ν A (x-y) max{ν A (x), ν A (y)} (ii) μ A (xn) μ A (x) and ν A (xn) ν A (x) (iii) μ A (y + x - y) μ A (x) and ν A (y + x - y) ν A (x) (iv) μ A (n( x + y)- nx) μ A (y) and ν A (n( x + y)- nx) ν A (y) Theorem (2.4) If A = (μ A, ν A ) be Intuitionistic fuzzy ideal of a near ring N, then (i) μ A (x) μ A (0) and ν A (x) ν A (0) for all x N (ii) μ A (- x ) = μ A (x) and ν A (- x ) = ν A (x) for all x N (iii) μ A (x + y) = μ A (y + x) and ν A (x + y) = ν A (y + x) for all x, y N Proof. Trivial Proof Proposition (2.5) If A = (μ A,ν A ) be Intuitionistic fuzzy ideal of a near ring N,then (i) μ A (x - y ) μ A ( 0 ) μ A ( x ) = μ A ( y ) (ii) ν A ( x- y ) ν A ( 0 ) ν A ( x ) = ν A ( y ) Proof. Trivial Proof Definition (2.6)[7] Let N and N be two near rings. Then the mapping f : N N is called (near ring) homomorphism if for all x, y N, the following holds (i) f ( x + y ) = f ( x ) + f ( y ) and (ii) f ( x y ) = f ( x ) f ( y ) Lemma ( 2.7) Let N and N be two near rings and let f : N N a near ring epimorphism Let 0 and 0 be additive identity element in N and N respectively such that f (0) = 0. If A = (μ A, ν A ) and B = (μ B, ν B ) are IFI s in N and N respectively. Then (i) μ f(a) ( 0 ) = μ A (0) and ν f(a) ( 0 ) = ν A (0) (ii) - μ f (B) ( 0 ) = μ B (0 ) and - ν f (B) ( 0 ) = ν B (0 ) Proof. Trivial Proof 3. ( α,β )-Cut of Intuitionistic fuzzy set (IFS) and their properties Definition (3.): ( α, β ) Cut of Intuitionistic fuzzy set Let A be Intuitionistic fuzzy set of a universe set X. Then ( α, β )-cut of A is a crisp subset C α, β (A) of the IFS A is given by C α, β (A) ={ x: x X s.t. μ A (x) α, ν A (x) β}, where α, β [0,] with α+β. Proposition (3.2)[ 9 ] If A and B be two IFS s of a universe set X, then
4 772 P. K. Sharma following holds (i) C α, β (A) C δ, θ(a) if α δ and β θ (ii) C -β, β (A) C α, β(a) C α, -α (A) (iii) A B implies C α, β (A) C α, β (B) (iv) C α, β (A B) = C α, β (A) C α, β (B) (v) C α, β (A B) C α, β (A) C α, β (B) equality hold if α + β = (vi) C α, β ( A i ) = C α, β (A i ) (vii) C 0, (A) = X. Proposition (3.3)[0] : Let f ; X Y be a mapping. Then the following holds (i) ( ) f C C ( f ( A )), A IFS( X ) α, β α, β (ii) ( ) f C ( B) = C ( f ( B )), B IFS( Y ) α, β α, β Proposition (3.4) If A = ( μ A, ν A ) be IFI in near ring N, then C α, β(a) is ideal of N if μ A (0) α, ν A (0) β Proof. Let μ A (0) α, ν A (0) β.. Clearly C α, β(a). Let x, y C α, β(a). Then μ A (x) α, ν A (x) β and μ A (y) α, ν A (y) β min{μ A (x), μ A (y) } α and max{ν A (x), ν A (y) } β Now μ A (x-y) min{μ A (x), μ A (y)} α and ν A (x-y) max{ ν A (x), ν A (y) } β μ A (x-y) α and ν A (x-y) β and so x y C α, β(a) Thus (C α, β(a), + ) is a subgroup of ( N, + ) To show that (C α, β(a), + ) is normal subgroup of ( N, + ) Let x C α, β(a) be any element and y N. we have μ A (x) α, ν A (x) β Since A is IFI of near ring N. μ A (y + x - y) μ A (x) α and ν A (y + x - y) ν A (x) β implies that y + x y C α, β(a). Thus (C α, β(a), + ) is normal subgroup of ( N, + ) Next to show that C α, β(a) N C α, β(a). Let x C α, β(a) and n N. Therefore we have μ A (x) α, ν A (x) β. As A is IFI of near ring N. μ A (xn) μ A (x) α and ν A (xn) ν A (x) β μ A (xn) α and ν A (xn) β And so xn C α, β(a). Thus C α, β(a) N C α, β(a) Next to show that n ( x + i ) nx C α, β(a) ; for all x, n N and i C α, β(a) Let i C α, β(a) be any element. Therefore we have μ A ( i ) α, ν A ( i ) β As A is IFI of near ring N, therefore we have μ A (n( x + i)- nx) μ A ( i ) α and ν A (n( x + i )- nx) ν A ( i ) β
5 Intuitionistic fuzzy ideals of near rings 773 i.e. μ A (n( x + i)- nx) α and ν A (n( x + i)- nx) β and so n( x + i )- nx C α, β(a) Hence C α, β(a) is Ideal in near ring N. Theorem (3.5) If A = ( μ A, ν A ) is an IFS of a near ring N, then A is IFI if and only if C α, β(a) is an ideal of N, for all α, β [0, ] with α + β and μ A (0) α, ν A (0) β. Proof If A be IFI of a near ring N, then C α, β(a) is an ideal of N, for all α, β [0,] with α + β and μ A (0) α, ν A (0) β follows from Proposition (3.4) Conversely, let A = ( μ A, ν A ) be IFS of a near ring N such that C α, β(a) is an ideal of N, for all α, β [0, ] with α + β and μ A (0) α, ν A (0) β. To show that A is IFI of near ring N. Let x, y N and α = min{ μ A (x), μ A (y)} and β = max{ ν A (x), ν A (y) } μ A (x) α, μ A (y) α and ν A (x) β, ν A (y) β μ A (x) α, ν A (x) β and μ A (y) α, ν A (y) β Therefore x, y C α, β(a). As C α, β(a) is ideal in near ring N x y C α, β(a) μ A (x - y) α = min{ μ A (x), μ A (y)} and ν A (x y) β = max{ ν A (x), ν A (y) } Thus μ A (x-y) min{μ A (x), μ A (y)} and ν A (x y) max{ν A (x), ν A (y) }.() As C α, β(a) N C α, β(a) holds for all α, β [0,] with α + β and μ A (0) α, ν A (0) β. Let x C α, β(a) be s.t μ A (x) = α and ν A (x) = β and n N be any element. Then xn C α, β(a) and so μ A (xn) α = μ A (x) and ν A (xn) β = ν A (x) i.e. μ A (xn) μ A (x) and ν A (xn) ν A (x) And if x C α, β(a) be such that μ A (x) = α and ν A (x) = β -α, where α α. Then x C A α, β ( ). As C α, β is Ideal in near ring N xn Cα, β μ A (xn) α = μ A (x) and ν A (xn) β = ν A (x) i.e. μ A (xn) μ A (x) and ν A (xn) ν A (x) Thus μ A (xn) μ A (x) and ν A (xn) ν A (x) holds for all x, n N (2) Next to show that μ A (y + x - y) μ A (x) and ν A (y + x - y) ν A (x) holds for all x, y N. As ( C α, β(a), + ) be normal subgroup of ( N, + ) Let x C α, β(a) be such that μ A (x) = α and ν A (x) = β and y N be any element. Then ( y + x y ) C α, β(a) μ A (y + x - y) α = μ A (x) and ν A (y + x - y) β = ν A (x) Now, if x C α, β(a) be s.t. μ A (x) = α and ν A (x) = β -α, where α α Then x Cα, β As C α, β is normal subgroup of N. So ( y+ x- y) C α, β μ A (y + x - y) α = μ A (x) and ν A (y + x - y) β = ν A (x)
6 774 P. K. Sharma i.e. μ A (y + x - y) μ A (x) and ν A (y + x - y) ν A (x) holds for all x, y N (3) Next to show that μ A (n( x + i)- nx) μ A (i) and ν A (n( x + i)- nx) ν A (i) holds for all x, n N, i A. Take i C α, β(a) be an element such that μ A ( i ) = α and ν A ( i ) = β. Since C α, β(a) is ideal of the near ring N Therefore, for x, n N, we have n( x + i)- nx C α, β(a) μ A (n( x + i)- nx) α = μ A (i) and ν A (n( x + i)- nx) β = ν A (i) and if i C α, β(a) be such that μ A (i) = α and ν A (i) = β -α, where α α Then i Cα, β As C α, β is ideal of N. n( x + i)- nx C α, β μ A (n( x + i)- nx) α = μ A (i) and ν A (n( x + i)- nx) β = ν A (i) i.e. μ A ( n (x + i)- nx) μ A (i) and ν A (n( x + i)- nx) ν A (i) (4) From (), (2), (3) and (4) we find that A is IFI of near ring N. Example (3.6) Let A = ( μ A, ν A ) be IFI of near ring N, then the set M = { x N : μ A (x) = μ A (0) and ν A (x) = ν A (0) } is ideal in near ring N. Proof. Easy to verify Corollary (3.7) Let N be near ring. Then the IFS A ={< x, μ A (x), ν A (x)>: x N : μ A (x) = μ A (0) and ν A (x) = ν A (0) } of N is IFI of the near ring N. Proof. Taking α = μ A (0) and β = ν A (0), then C α, β(a) = M Therefore by Theorem (3.5), A is IFI of near ring N. Theorem (3.8) If A and B be two IFI s of a near ring N, then A B is also IFI of N. Proof. Since A and B be two IFI s of near ring N. By Proposition (3.4), we have C α, β(a) and C α, β(b) are ideals in near ring N. Since intersection of two ideals in near ring is ideal in N. Therefore C α, β(a) C α, β(b) is ideal in N C α, β(a B) is ideal in N (by using Proposition 3.2(iv)) A B is IFI in near ring N ( by Theorem (3.5)converse part ) Corollary (3.9) Intersection of a family of IFI s of near ring N is IFI in N Theorem (3.0) Let N and N be two near rings and let f : N N be near ring homomorphism. If B = ( μ B, ν B ) is an IFI in N, then the pre-image f - ( B) of B under f is an IFI of N. Proof. Since B is IFI in near ring N C α, β(b) is ideal in N, for all α, β [0,] with α + β and μ B (0) α, ν B (0) β [ By Proposition (3.4) ]. f - (C α, β(b)) is ideal in N. But f - (C α, β(b)) = C α, β( f - (B))[By Theorem(3.3)] - C α, β( f ( B)) is ideal in N and by using Theorem (3.5), we get f - ( B) is IFI in near ring N.
7 Intuitionistic fuzzy ideals of near rings 775 Theorem (3.) Let N and N be two near rings and let f : N N be epimorphism If A = ( μ A, ν A ) is IFI of N, then f is IFI of N. Proof. In view of Theorem (3.5), it is enough to show that C α, β( f ) is ideal in N, for all α, β [0,] with α + β and μ f(a) (0 ) α, ν f(a) (0 ) β. Let y, y 2 C α, β( f (A)) be any two elements, then μ ( )( y) α, ν A ( y) β and μ ( y2) α, ν ( y2) β f f f f By Proposition (3.3)(i), we have f C C ( f ( A )), A IFS( N) ( ) α, β α, β Therefore s x and x 2 in N such that f (x ) = y, f ( x 2 ) = y 2 and μ ( x ) μ ( y ) α, ν ( x ) ν ( y ) β and μ ( x ) μ ( y ) α, ν ( x ) ν ( y ) β A f A f f 2 f 2 A 2 f 2 μ( x) α, ν ( x) β and μ( x) α, ν ( x) β A A A 2 A 2 A x A x2 A x A x2 min{ μ ( ), μ ( )} α and max { ν ( ), ν ( )} β As A is IFI of near ring N. Therefore μa( x x2) min{ μa( x), μa( x2)} α and νa( x x2) max{ νa( x), νa( x2)} β μ ( x x ) α and ν ( x x ) β A 2 A 2 ( ) ( ) x x C f x x f C C ( f ) 2 α, β 2 α, β α, β f ( x ) f ( x2 ) Cα, β( f ) y- y 2 Cα, β( f ) Hence ( C ( f ), + ) is a subgroup of ( N, +) α, β Next to show that ( C, ( f ), + ) is normal subgroup of ( N, +) Let y C α, β( f (A)) and n be any elements, then as above s x and n in N such that f (x ) = y, f ( n ) = n and μ ( x ) μ ( y ) α, ν ( x ) ν ( y ) β μ ( x ) α, ν ( x ) β α β A f A f A A x C α, β(a). As A is IFI of N, therefore C α, β(a) is ideal in N ( n + x n ) C α, β(a) f (( n + x n )) f (C α, β(a)) C α, β ( f(a)) f ( n ) + ( x ) ( n ) C α, β ( f(a)) i.e. ( n + y - n ) C α, β ( f(a)) Thus ( C ( f ), + ) is normal subgroup of ( N, +) α, β Now to show that C α, β(f (A))N N. As above, we get x n N f (x n ) = f (x ) f (n) = y n f ( N ) = N Further, to show that n ( y + y 2 ) - n y C α, β( f (A)) for all y, y 2, n N. As above s x and x 2 in N such that f (x ) = y, f (x 2 ) = y 2 and f(n) = n such that x, x 2 C α, β(a). As C α, β(a) is ideal in N. Therefore ( x + x 2 ) - n x C α, β(a) f ( n ( x + x 2 ) - n x ) f ( C α, β(a)) C α, β ( f(a)) f ( n ) ( f ( x ) + f ( x 2 )) f ( n) f ( x ) = n ( y + y 2 ) - n y C α, β( f (A))
8 776 P. K. Sharma Hence f (A) is IFI in near ring N. References K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and systems 20(986) p D.K. Basnet, (α,β) cut of intuitionistic fuzzy ideals,nifs,6(200),, R. Biswas, Intuitionistic fuzzy subgroups, Math Forum 0 (987) p Y.H.Yon, Y.B. Jun and K.H. Kim Intuitionistic fuzzy R-subgroups of near-rings, Soochow Journal of Mathematics, Vol. 27, No. 3, (200), p S.M. Hong, Y.B. Jun, H.S. Kim, Fuzzy ideals in near-rings, Bull of Korean Math Soc. 35(3) (998) p O. Kazanci, S. Yamak, S. Yilmaz, On Intuitionistic Q-fuzzy R-subgroups of near-rings, International Math Forum, 2, 2007(59) G. Pilz, Near-Rings, volume 23 of North-Holland Mathematics Studies. North-Holland, Amsterdam, 2nd edition, O. Ratnabala Devi, On the Intuitionistic Q-fuzzy ideals of near rings, NIFS 5 (2009), 3, P.K. Sharma, (α, β) -Cut of Intuitionistic fuzzy groups, International Mathematics Forum, Vol. 6, 20, no. 53, P.K. Sharma, Homomorphism of Intuitionistic fuzzy groups, International Mathematics Forum, Vol. 6, 20, no. 64, Y.H. Yon, Y.B. Jun, K.H. Kim, Intuitionistic fuzzy R-subgroups of near-rings, Shoochow J. Math, 27(No.3)(200) p L.A. Zadeh, Fuzzy Sets, Information and Controls, Vol. 83(965), p Zhan Jianming, Ma Xueling, Intuitionistic fuzzy ideals of near-rings, Scientae Math Japonicae, 6 (No.2), (2004)p Received: September, 20
Homomorphism of Intuitionistic Fuzzy Groups
International Mathematical Forum, Vol. 6, 20, no. 64, 369-378 Homomorphism o Intuitionistic Fuzz Groups P. K. Sharma Department o Mathematics, D..V. College Jalandhar Cit, Punjab, India pksharma@davjalandhar.com
Διαβάστε περισσότεραA Note on Intuitionistic Fuzzy. Equivalence Relation
International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com
Διαβάστε περισσότεραHomomorphism in Intuitionistic Fuzzy Automata
International Journal of Fuzzy Mathematics Systems. ISSN 2248-9940 Volume 3, Number 1 (2013), pp. 39-45 Research India Publications http://www.ripublication.com/ijfms.htm Homomorphism in Intuitionistic
Διαβάστε περισσότεραA Note on Characterization of Intuitionistic Fuzzy Ideals in Γ- Near-Rings
International Journal of Computational Science and Mathematics. ISSN 0974-3189 Volume 3, Number 1 (2011), pp. 61-71 International Research Publication House http://www.irphouse.com A Note on Characterization
Διαβάστε περισσότεραCommutative Monoids in Intuitionistic Fuzzy Sets
Commutative Monoids in Intuitionistic Fuzzy Sets S K Mala #1, Dr. MM Shanmugapriya *2 1 PhD Scholar in Mathematics, Karpagam University, Coimbatore, Tamilnadu- 641021 Assistant Professor of Mathematics,
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραHomomorphism and Cartesian Product on Fuzzy Translation and Fuzzy Multiplication of PS-algebras
Annals of Pure and Applied athematics Vol. 8, No. 1, 2014, 93-104 ISSN: 2279-087X (P), 2279-0888(online) Published on 11 November 2014 www.researchmathsci.org Annals of Homomorphism and Cartesian Product
Διαβάστε περισσότεραGÖKHAN ÇUVALCIOĞLU, KRASSIMIR T. ATANASSOV, AND SINEM TARSUSLU(YILMAZ)
IFSCOM016 1 Proceeding Book No. 1 pp. 155-161 (016) ISBN: 978-975-6900-54-3 SOME RESULTS ON S α,β AND T α,β INTUITIONISTIC FUZZY MODAL OPERATORS GÖKHAN ÇUVALCIOĞLU, KRASSIMIR T. ATANASSOV, AND SINEM TARSUSLU(YILMAZ)
Διαβάστε περισσότεραGenerating Set of the Complete Semigroups of Binary Relations
Applied Mathematics 06 7 98-07 Published Online January 06 in SciRes http://wwwscirporg/journal/am http://dxdoiorg/036/am067009 Generating Set of the Complete Semigroups of Binary Relations Yasha iasamidze
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότεραCongruence Classes of Invertible Matrices of Order 3 over F 2
International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and
Διαβάστε περισσότεραSOME PROPERTIES OF FUZZY REAL NUMBERS
Sahand Communications in Mathematical Analysis (SCMA) Vol. 3 No. 1 (2016), 21-27 http://scma.maragheh.ac.ir SOME PROPERTIES OF FUZZY REAL NUMBERS BAYAZ DARABY 1 AND JAVAD JAFARI 2 Abstract. In the mathematical
Διαβάστε περισσότεραCHARACTERIZATION OF BIPOLAR FUZZY IDEALS IN ORDERED GAMMA SEMIGROUPS
JOURNAL OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4866, ISSN (o) 2303-4947 www.imvibl.org /JOURNALS / JOURNAL Vol. 8(2018), 141-156 DOI: 10.7251/JIMVI1801141C Former BULLETIN OF
Διαβάστε περισσότεραDIRECT PRODUCT AND WREATH PRODUCT OF TRANSFORMATION SEMIGROUPS
GANIT J. Bangladesh Math. oc. IN 606-694) 0) -7 DIRECT PRODUCT AND WREATH PRODUCT OF TRANFORMATION EMIGROUP ubrata Majumdar, * Kalyan Kumar Dey and Mohd. Altab Hossain Department of Mathematics University
Διαβάστε περισσότεραTHE SECOND ISOMORPHISM THEOREM ON ORDERED SET UNDER ANTIORDERS. Daniel A. Romano
235 Kragujevac J. Math. 30 (2007) 235 242. THE SECOND ISOMORPHISM THEOREM ON ORDERED SET UNDER ANTIORDERS Daniel A. Romano Department of Mathematics and Informatics, Banja Luka University, Mladena Stojanovića
Διαβάστε περισσότερα1. Introduction and Preliminaries.
Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.yu/filomat Filomat 22:1 (2008), 97 106 ON δ SETS IN γ SPACES V. Renuka Devi and D. Sivaraj Abstract We
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραUniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Διαβάστε περισσότεραSOME INTUITIONISTIC FUZZY MODAL OPERATORS OVER INTUITIONISTIC FUZZY IDEALS AND GROUPS
IFSCOM016 1 Proceeding Book No. 1 pp. 84-90 (016) ISBN: 978-975-6900-54-3 SOME INTUITIONISTIC FUZZY MODAL OPERATORS OVER INTUITIONISTIC FUZZY IDEALS AND GROUPS SINEM TARSUSLU(YILMAZ), GÖKHAN ÇUVALCIOĞLU,
Διαβάστε περισσότεραNowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραSome new generalized topologies via hereditary classes. Key Words:hereditary generalized topological space, A κ(h,µ)-sets, κµ -topology.
Bol. Soc. Paran. Mat. (3s.) v. 30 2 (2012): 71 77. c SPM ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v30i2.13793 Some new generalized topologies via hereditary
Διαβάστε περισσότεραSCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Διαβάστε περισσότεραCoefficient Inequalities for a New Subclass of K-uniformly Convex Functions
International Journal of Computational Science and Mathematics. ISSN 0974-89 Volume, Number (00), pp. 67--75 International Research Publication House http://www.irphouse.com Coefficient Inequalities for
Διαβάστε περισσότεραF A S C I C U L I M A T H E M A T I C I
F A S C I C U L I M A T H E M A T I C I Nr 46 2011 C. Carpintero, N. Rajesh and E. Rosas ON A CLASS OF (γ, γ )-PREOPEN SETS IN A TOPOLOGICAL SPACE Abstract. In this paper we have introduced the concept
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Διαβάστε περισσότεραORDINAL ARITHMETIC JULIAN J. SCHLÖDER
ORDINAL ARITHMETIC JULIAN J. SCHLÖDER Abstract. We define ordinal arithmetic and show laws of Left- Monotonicity, Associativity, Distributivity, some minor related properties and the Cantor Normal Form.
Διαβάστε περισσότεραMINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS
MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS FUMIE NAKAOKA AND NOBUYUKI ODA Received 20 December 2005; Revised 28 May 2006; Accepted 6 August 2006 Some properties of minimal closed sets and maximal closed
Διαβάστε περισσότεραk A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Διαβάστε περισσότεραOn Annihilator of Fuzzy Subsets of Modules
International Journal of Algebra, Vol. 3, 2009, no. 10, 483-488 On Annihilator of Fuzzy Subsets of Modules Helen K. Saikia 1 and Mrinal C. Kalita 2 1 Department of Mathematics, Gauhati university, Guwahati-781014,
Διαβάστε περισσότεραLecture 15 - Root System Axiomatics
Lecture 15 - Root System Axiomatics Nov 1, 01 In this lecture we examine root systems from an axiomatic point of view. 1 Reflections If v R n, then it determines a hyperplane, denoted P v, through the
Διαβάστε περισσότεραThe operators over the generalized intuitionistic fuzzy sets
Int. J. Nonlinear Anal. Appl. 8 (2017) No. 1, 11-21 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2017.11099.1542 The operators over the generalized intuitionistic fuzzy sets Ezzatallah
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραA General Note on δ-quasi Monotone and Increasing Sequence
International Mathematical Forum, 4, 2009, no. 3, 143-149 A General Note on δ-quasi Monotone and Increasing Sequence Santosh Kr. Saxena H. N. 419, Jawaharpuri, Badaun, U.P., India Presently working in
Διαβάστε περισσότεραSEMI DERIVATIONS OF PRIME GAMMA RINGS
GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) xx (2011) 31 (2011) 65-70 SEMI DERIVATIONS OF PRIME GAMMA RINGS Kalyan Kumar Dey 1 and Akhil Chandra Paul 2 1,2 Department of Mathematics Rajshahi University,
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραLecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Διαβάστε περισσότεραOn Intuitionistic Fuzzy LI -ideals in Lattice Implication Algebras
Journal of Mathematical Research with Applications Jul., 2015, Vol. 35, No. 4, pp. 355 367 DOI:10.3770/j.issn:2095-2651.2015.04.001 Http://jmre.dlut.edu.cn On Intuitionistic Fuzzy LI -ideals in Lattice
Διαβάστε περισσότεραYoung Bae Jun Madad Khan Florentin Smarandache Saima Anis. Fuzzy and Neutrosophic Sets in Semigroups
Young Bae Jun Madad Khan Florentin Smarandache Saima Anis Fuzzy and Neutrosophic Sets in Semigroups Young Bae Jun, Madad Khan, Florentin Smarandache, Saima Anis Fuzzy and Neutrosophic Sets in Semigroups
Διαβάστε περισσότεραHomework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότεραMath 446 Homework 3 Solutions. (1). (i): Reverse triangle inequality for metrics: Let (X, d) be a metric space and let x, y, z X.
Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequalit for metrics: Let (X, d) be a metric space and let x,, z X. Prove that d(x, z) d(, z) d(x, ). (ii): Reverse triangle inequalit for norms:
Διαβάστε περισσότεραFuzzy Soft Rings on Fuzzy Lattices
International Journal of Computational Science and Mathematics. ISSN 0974-389 Volume 3, Number 2 (20), pp. 4-59 International Research Publication House http://www.irphouse.com Fuzzy Soft Rings on Fuzzy
Διαβάστε περισσότεραProblem Set 3: Solutions
CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότερα2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2005-03-08 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
Διαβάστε περισσότεραOperation Approaches on α-γ-open Sets in Topological Spaces
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 10, 491-498 Operation Approaches on α-γ-open Sets in Topological Spaces N. Kalaivani Department of Mathematics VelTech HighTec Dr.Rangarajan Dr.Sakunthala
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραPractice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
Διαβάστε περισσότεραAbstract Storage Devices
Abstract Storage Devices Robert König Ueli Maurer Stefano Tessaro SOFSEM 2009 January 27, 2009 Outline 1. Motivation: Storage Devices 2. Abstract Storage Devices (ASD s) 3. Reducibility 4. Factoring ASD
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραThe strong semilattice of π-groups
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 3, 2018, 589-597 ISSN 1307-5543 www.ejpam.com Published by New York Business Global The strong semilattice of π-groups Jiangang Zhang 1,, Yuhui
Διαβάστε περισσότεραChapter 3: Ordinal Numbers
Chapter 3: Ordinal Numbers There are two kinds of number.. Ordinal numbers (0th), st, 2nd, 3rd, 4th, 5th,..., ω, ω +,... ω2, ω2+,... ω 2... answers to the question What position is... in a sequence? What
Διαβάστε περισσότεραIntuitionistic Supra Gradation of Openness
Applied Mathematics & Information Sciences 2(3) (2008), 291-307 An International Journal c 2008 Dixie W Publishing Corporation, U. S. A. Intuitionistic Supra Gradation of Openness A. M. Zahran 1, S. E.
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραSequent Calculi for the Modal µ-calculus over S5. Luca Alberucci, University of Berne. Logic Colloquium Berne, July 4th 2008
Sequent Calculi for the Modal µ-calculus over S5 Luca Alberucci, University of Berne Logic Colloquium Berne, July 4th 2008 Introduction Koz: Axiomatisation for the modal µ-calculus over K Axioms: All classical
Διαβάστε περισσότεραb. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
Διαβάστε περισσότεραSolutions to Selected Homework Problems 1.26 Claim: α : S S which is 1-1 but not onto β : S S which is onto but not 1-1. y z = z y y, z S.
Solutions to Selected Homework Problems 1.26 Claim: α : S S which is 1-1 but not onto β : S S which is onto but not 1-1. Proof. ( ) Since α is 1-1, β : S S such that β α = id S. Since β α = id S is onto,
Διαβάστε περισσότεραOn a four-dimensional hyperbolic manifold with finite volume
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραFractional Colorings and Zykov Products of graphs
Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is
Διαβάστε περισσότεραw o = R 1 p. (1) R = p =. = 1
Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότεραFuzzifying Tritopological Spaces
International Mathematical Forum, Vol., 08, no. 9, 7-6 HIKARI Ltd, www.m-hikari.com https://doi.org/0.988/imf.08.88 On α-continuity and α-openness in Fuzzifying Tritopological Spaces Barah M. Sulaiman
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραJordan Journal of Mathematics and Statistics (JJMS) 4(2), 2011, pp
Jordan Journal of Mathematics and Statistics (JJMS) 4(2), 2011, pp.115-126. α, β, γ ORTHOGONALITY ABDALLA TALLAFHA Abstract. Orthogonality in inner product spaces can be expresed using the notion of norms.
Διαβάστε περισσότεραSection 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Διαβάστε περισσότεραF19MC2 Solutions 9 Complex Analysis
F9MC Solutions 9 Complex Analysis. (i) Let f(z) = eaz +z. Then f is ifferentiable except at z = ±i an so by Cauchy s Resiue Theorem e az z = πi[res(f,i)+res(f, i)]. +z C(,) Since + has zeros of orer at
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότερα5. Choice under Uncertainty
5. Choice under Uncertainty Daisuke Oyama Microeconomics I May 23, 2018 Formulations von Neumann-Morgenstern (1944/1947) X: Set of prizes Π: Set of probability distributions on X : Preference relation
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότεραCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Διαβάστε περισσότεραSEMIGROUP ACTIONS ON ORDERED GROUPOIDS. 1. Introduction and prerequisites
ao DOI: 10.2478/s12175-012-0080-3 Math. Slovaca 63 (2013), No. 1, 41 52 SEMIGROUP ACTIONS ON ORDERED GROUPOIDS Niovi Kehayopulu* Michael Tsingelis** (Communicated by Miroslav Ploščica ) ABSTRACT. In this
Διαβάστε περισσότεραΑπόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Διαβάστε περισσότεραLecture 13 - Root Space Decomposition II
Lecture 13 - Root Space Decomposition II October 18, 2012 1 Review First let us recall the situation. Let g be a simple algebra, with maximal toral subalgebra h (which we are calling a CSA, or Cartan Subalgebra).
Διαβάστε περισσότεραLecture 21: Properties and robustness of LSE
Lecture 21: Properties and robustness of LSE BLUE: Robustness of LSE against normality We now study properties of l τ β and σ 2 under assumption A2, i.e., without the normality assumption on ε. From Theorem
Διαβάστε περισσότεραMATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)
1. MATH43 String Theory Solutions 4 x = 0 τ = fs). 1) = = f s) ) x = x [f s)] + f s) 3) equation of motion is x = 0 if an only if f s) = 0 i.e. fs) = As + B with A, B constants. i.e. allowe reparametrisations
Διαβάστε περισσότεραFourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραQuadratic Expressions
Quadratic Expressions. The standard form of a quadratic equation is ax + bx + c = 0 where a, b, c R and a 0. The roots of ax + bx + c = 0 are b ± b a 4ac. 3. For the equation ax +bx+c = 0, sum of the roots
Διαβάστε περισσότεραThe Translational Hulls of Inverse wpp Semigroups 1
International Mathematical Forum, 4, 2009, no. 28, 1397-1404 The Translational Hulls of Inverse wpp Semigroups 1 Zhibin Hu Department of Mathematics, Jiangxi Normal University Nanchang, Jiangxi 330022,
Διαβάστε περισσότεραStrain gauge and rosettes
Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified
Διαβάστε περισσότεραand Institute for Mathematical Research (INSPEM) Universiti Putra Malaysia MALAYSIA 2,3 Department of Mathematics
International Journal of Pure and Applied Mathematics Volume 82 No. 5 2013, 669-681 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v82i5.1
Διαβάστε περισσότεραThe k-α-exponential Function
Int Journal of Math Analysis, Vol 7, 213, no 11, 535-542 The --Exponential Function Luciano L Luque and Rubén A Cerutti Faculty of Exact Sciences National University of Nordeste Av Libertad 554 34 Corrientes,
Διαβάστε περισσότεραSecond Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Διαβάστε περισσότερα= λ 1 1 e. = λ 1 =12. has the properties e 1. e 3,V(Y
Stat 50 Homework Solutions Spring 005. (a λ λ λ 44 (b trace( λ + λ + λ 0 (c V (e x e e λ e e λ e (λ e by definition, the eigenvector e has the properties e λ e and e e. (d λ e e + λ e e + λ e e 8 6 4 4
Διαβάστε περισσότεραNew Operations over Interval Valued Intuitionistic Hesitant Fuzzy Set
Mathematics and Statistics (): 6-7 04 DOI: 0.89/ms.04.000 http://www.hrpub.org New Operations over Interval Valued Intuitionistic Hesitant Fuzzy Set Said Broumi * Florentin Smarandache Faculty of Arts
Διαβάστε περισσότεραMINIMAL INTUITIONISTIC GENERAL L-FUZZY AUTOMATA
italian journal of pure applied mathematics n. 35 2015 (155 186) 155 MINIMAL INTUITIONISTIC GENERAL L-UZZY AUTOMATA M. Shamsizadeh M.M. Zahedi Department of Mathematics Kerman Graduate University of Advanced
Διαβάστε περισσότεραChapter 2. Ordinals, well-founded relations.
Chapter 2. Ordinals, well-founded relations. 2.1. Well-founded Relations. We start with some definitions and rapidly reach the notion of a well-ordered set. Definition. For any X and any binary relation
Διαβάστε περισσότεραOn a Subclass of k-uniformly Convex Functions with Negative Coefficients
International Mathematical Forum, 1, 2006, no. 34, 1677-1689 On a Subclass of k-uniformly Convex Functions with Negative Coefficients T. N. SHANMUGAM Department of Mathematics Anna University, Chennai-600
Διαβάστε περισσότεραAffine Weyl Groups. Gabriele Nebe. Summerschool GRK 1632, September Lehrstuhl D für Mathematik
Affine Weyl Groups Gabriele Nebe Lehrstuhl D für Mathematik Summerschool GRK 1632, September 2015 Crystallographic root systems. Definition A crystallographic root system Φ is a finite set of non zero
Διαβάστε περισσότερα