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2 Vol. 8, No. 4, 01 ISSN SCIENTIA MAGNA An international journal Edited by Department of Mathematics Northwest University Xi an, Shaanxi, P.R.China

3 Scientia Magna is published annually in pages per volume and 1,000 copies. It is also available in microfilm format and can be ordered (online too from: Books on Demand ProQuest Information & Learning 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Michigan , USA Tel.: (Customer Service URL: Scientia Magna is a referred journal: reviewed, indexed, cited by the following journals: "Zentralblatt Für Mathematik" (Germany, "Referativnyi Zhurnal" and "Matematika" (Academia Nauk, Russia, "Mathematical Reviews" (USA, "Computing Review" (USA, Institute for Scientific Information (PA, USA, "Library of Congress Subject Headings" (USA. Printed in the United States of America Price: US$ i

4 Information for Authors Papers in electronic form are accepted. They can be ed in Microsoft Word XP (or lower, WordPerfect 7.0 (or lower, LaTeX and PDF 6.0 or lower. The submitted manuscripts may be in the format of remarks, conjectures, solved/unsolved or open new proposed problems, notes, articles, miscellaneous, etc. They must be original work and camera ready [typewritten/computerized, format: 8.5 x 11 inches (1,6 x 8 cm]. They are not returned, hence we advise the authors to keep a copy. The title of the paper should be writing with capital letters. The author's name has to apply in the middle of the line, near the title. References should be mentioned in the text by a number in square brackets and should be listed alphabetically. Current address followed by address should apply at the end of the paper, after the references. The paper should have at the beginning an abstract, followed by the keywords. All manuscripts are subject to anonymous review by three independent reviewers. Every letter will be answered. Each author will receive a free copy of the journal. ii

5 Contributing to Scientia Magna Authors of papers in science (mathematics, physics, philosophy, psychology, sociology, linguistics should submit manuscripts, by , to the Editor-in-Chief: Prof. Wenpeng Zhang Department of Mathematics Northwest University Xi an, Shaanxi, P.R.China Or anyone of the members of Editorial Board: Dr. W. B. Vasantha Kandasamy, Department of Mathematics, Indian Institute of Technology, IIT Madras, Chennai , Tamil Nadu, India. Dr. Larissa Borissova and Dmitri Rabounski, Sirenevi boulevard , Moscow , Russia. Prof. Yuan Yi, Research Center for Basic Science, Xi an Jiaotong University, Xi an, Shaanxi, P.R.China. yuanyi@mail.xjtu.edu.cn Prof. Zhefeng Xu, Department of Mathematics, Northwest University, Xi an, Shaanxi, P.R.China. zfxu@nwu.edu.cn; zhefengxu@hotmail.com Prof. József Sándor, Babes-Bolyai University of Cluj, Romania. jjsandor@hotmail.com; jsandor@member.ams.org Dr. Chenglong Li, Department of Mathematics, Northwest University, Xi an, Shaanxi, P.R.China. david1739@16.com Dr. Haimei Gao, Department of Mathematics, Northwest University, Xi an, Shaanxi, P.R.China. hmgaomoveon@163.com iii

6 Contents M. Dragan and M. Bencze : A new refinement of the inequality sin A 4R+r r 1 S. S. Billing : On certain subclasses of analytic functions 11 Salahuddin : Certain indedinite integral associate to elliptic integral and complex argument 17 A. C. F. Bueno : On the Types and Sums of Self-Intergrating Functions 5 S. Uddin : Creation of a summation formula enmeshed with contiguous relation 7 V. Maheswari and A. Nagarajan : Distar decompositions of a symmetric product digraph 43 N. Subramanian, etc. : The generalized difference gai sequences of fuzzy numbers defined by Orlicz functions 51 G. Singh : Hankel Determinant for a new subclass 61 Y. Unluturk, etc. : On non-unit speed curves in Minkowski 3-space 66 A. C. F. Bueno : Self-integrating Polynomials in two variables 75 P. Lawrence and R. Lawrence : Signed product cordial graphs in the context of arbitrary supersubdivision 77 A. C. F. Bueno : Smarandache Cyclic Geometric Determinant Sequences 88 A. A. Mogbademu : Some convergence results for asymptotically generalized Φ-hemicontractive mappings 9 N. Subramanian, etc. : The v invariant χ sequence spaces 101 P. Rajarajeswari, etc. : Weakly generalized compactness in intuitionistic fuzzy topological spaces 108 W. Wang, etc. : On normal and fantastic filters of BL-algebras 118 iv

7 Scientia Magna Vol. 8 (01, No. 4, 1-10 A new refinement of the inequality sin A 4R+r r Marius Dragan and Mihály Bencze benczemihaly@gmail.com Abstract The purpose of this paper is to give a new provement to inequality sin A 4R+r r, who are given in [1], to prove that this is the better inequality of type: sin A αr+βr R, when αr + βr R 3, (1 and to refine this inequality with an equality of type: better of the type : when or in an equivalent from: sin A αr + βr R αr + βr, R 3 ( We denote ( A R + 3 r sin. (3 R R r = x, d = R Rr, d x = x x. Keywords Geometrical inequalities. 010 Mathematics Subject Classification: 6D15, 6D15, 51M Main results Lemma 1.1. In all triangle ABC holds: R + ( 3 r Proof. The inequality (4 will be written as: R + 3 x 4R + r R. (4 4x + 1 x

8 Marius Dragan and Mihály Bencze No. 4 and after squaring we shall obtain: x x ( x ( 17 1 (x 0. Theorem 1.1. In all triangle ABC holds: sin A r R + d + 4x + 1 x x 4x + x R + d R. (5 Proof. The inequality (5 will be given in []. We shall give a new prove of this inequality. In [3] it was proved the following inequality: We have: p a a R r + d R d + r R. ( sin A or in an equivalent form: = ( sin A + sin B sin A = (p b (p c bc = = R r + sin C (p a (p b (p c abc r p a R a ( r R r + d R r R + R r R d R + + R R r R r + d R + + r (R d R R p a a, and because and (R + d = R (R r + d (R d (R + d = Rr (R r d (R r + d = r, it shall result sin A = R + d R + R r R r R + d + In the following we shall prove (1. R + d R. R + d R + r R + d R + d R

9 Vol. 8 A new refinement of the inequality sin A 4R+r r 3 From inequality (5: sin A + sin B + sin C R + d R + r R + d 3, because the inequality (5 is the better of the type: sin A f (R, r. It follows that: R + d R + 1 αr + βr R + d 3 R or 1 x + dx αx + β + 3, x. (6 1 + d x x x In the case of equilateral triangle we have x =. We shall obtain 4α + 3 = 9. The inequality (6 may be written in the case of the isosceles triangle with sides: b c = 1, a = 0 (R = 1, r = 0 or putting x in an equivalent from as: a. Because: (α 4 x + 4x + 9 4α (α 4 + 4x + 9 4α = 4x + 1 it shall result: αx + β x 4x + 1 x. In the following will be sufficient to prove that: x + dx Theorem 1.. In all triangle ABC holds x + 1 4x + 1 x + d x x. (7 A 4R + r sin. r Proof. The inequality (7 may be written in an equivalent form as: x + dx x + 1 4x + 1 x + d x λ 1 4x + 1 x + dx x + d x x x 4x + 1 x + dx + x x ( 4x + 1 x dx (x + d x x = d x + 5x x d x + 5x x (4x x (x + d x.

10 4 Marius Dragan and Mihály Bencze No. 4 After squaring we shall obtain: x x + 5x + 10xd x 8x + x + 4x + 4xd x + 4x (4x + 1 (x + d x 14x 4x + 6xdx 4x (4x + 1 (x + d x 7x + 3d x (8x + (x + d x 49x x 18x 8x 1d x + 4xd x 3x + 3xd x + 8x + 8d x 6x 54x xd x 0d x 0 (13x 1 + 5d x (x 0. In the following we shall determine α, β with the property (. According with the inequality (6 it follows that or in an equivalent form R + d R + r αr + βr 3 R + d R x + dx x In the case of equilateral triangle we shall obtain α + β = x + dx α + β x 3. (8 In the inequality (8 we shall consider x. It follows that a. Because ( α ( R + (3 α r α ( + 3 α r = 3 r, it follows that ( αr + βr R + 3 r. R R In the following will be sufficient to prove the inequality (3. Theorem 1.3. In all triangle ABC holds ( A R + 3 r sin. R Proof. From the inequality (5 it follows that in order to prove the inequality (3 it will be sufficient to prove: R + d R + r ( R = 3 r R + d R or in an equivalent form: x + dx + 1 x + 3 x x + dx x x + dx x + ( 3 x + xd x + ( 3 d x x x x (x + d x

11 Vol. 8 A new refinement of the inequality sin A 4R+r r 5 x + dx x + ( x x + xd x + ( 3 d x. (9 x x (x + d x We denote u x = d x x. The inequality (9 may be written in an equivalent form as: x + d x x [ x x + + d x + ( 3 ] u x x 3 (x + d x 1 (x + d x (x + d x 1 [ ( x + + dx + 3 ] u x x + xd x x + xd x + x 4x x x ( + x 4x u x ( + 4 ( 8 x + 4xd x + 6 ( 8 xu x ( ( u x + 4 ( 8 d x u x d x 4xd x 6x d x 1 8 (x ( ( 4x x x ( + 6 ( 8 d x + 4xd x dx ( x + 4 ( 8 d x + (x 6 8 4x d x 6x xd ( x 1 8 ( x x x ( + 4 ( 8 x + 6 ( 8 xd x + 4x d x d x ( + 4 ( 8 xd x + 6 ( 8 x 1 16 x [ ( 4x x 6 8 x 4x ( 4 8 ( x ( x d x [( x ] ( 10 ( 14 x x d x [(14 10 x + 0 ] 8 ( 10 ( 14 x x d x [(14 10 x + 0 ] 8 ( 10 ( 14 x x d x (14 10 [( (x (x d x ] x d x x ]

12 6 Marius Dragan and Mihály Bencze No. 4 [( (x 10 ( 14 x d x ] 0. Corollary 1.1. In all triangle ABC holds cos A 3S ( R + ( 3 r. Proof. Using the Chebyshev s inequality to sin A, sin B, sin C and cos A, cos B, cos C we get ( 1 3 or or ( A 1 sin 3 ( A 1 A cos sin 3 cos A A A sin cos 3 sin A 3 sin A cos + 3 sin AR sin A [ R + ( 3 r ] = 3S ( R + ( 3 r. Corollary 1.. If λ (0, 1], then in all triangle ABC holds: ( sin A 3R λ 3 ( ( λ R + 3 r. Proof. Using the Jensen s inequality we get: ( sin A λ ( λ A sin = 3 Corollary 1.3. If λ 1 then in all triangle ABC holds ( ( λ R + 3 r. 3R ( cos A ( λ λ S 3 ( R + ( 3 r. Proof. From Jensen s inequality we get: ( cos A λ ( cos A ( λ λ S 3 ( R + ( 3 r. Corollary 1.4. In all triangle ABC holds A cos S + 3 3R. 4R Proof. From Popoviciu s inequality f (x + 3f ( x + y + z 3 ( x + y f applied to function f : (0, π R, f (x = sin x we get: π sin A + 3 sin 3 sin A + B.

13 Vol. 8 A new refinement of the inequality sin A 4R+r r 7 or or S R cos A A cos S + 3 3R. 4R Corollary 1.5. In all triangle ABC are true the following equality: R cos A ( A sin 1 = S. (10 Proof. We shall consider the triangle ABC obtained with the exterior bisectors of the triangle ABC. We denote with S A1 B 1 C 1 the semiperimeter of A 1 B 1 C 1 trianle. We have: S A1 B 1 C 1 = A 1 B 1 + B 1 C 1 + A 1 C 1. We shall calculate: B 1 C 1 = AB 1 + AC 1 = 4R sin C cos B + 4R sin B cos C = 4R cos A and the others. We shall obtain: S A1 B 1 C 1 = R cos A. We have: S ABC1 + S BCA1 + S ACB1 = S ABC + S A1 B 1 C 1 and S ABC1 = 4R sin C cos B + 4R sin C cos C = 4R sin C A cos.

14 8 Marius Dragan and Mihály Bencze No. 4 It follows that ( A R cos sin C ( + R A cos sin B ( + R A cos sin A = S + R cos A or R cos A Corollary 1.6. In all triangle ABC holds: ( A sin 1 = S. A cos S [( 1 R + ( 3 ]. (11 r Proof. Using the equality (10 and inequality (3 we shall obtain: cos A = S R = 1 sin A 1 S R S [( 1 R + ( 3 r ]. R ( 1 R + ( 3 r Corollary 1.7. If λ 1 then in all triangle ABC holds: ( cos A ( λ λ S 3 6 (( 1 R + ( 3 r. Proof. From Jensen s inequality and inequality (11 we get: ( cos A λ ( ( λ A cos 3 Corollary 1.8. In all triangle ABC holds: S 6 (( 1 R + ( 3 r λ. A cos S + 3 3R 4R 3S ( R + ( 3. (1 r Proof. The first side of the inequality (1 i just Corollary 3.4. The right side S + 3 3R 3S 4R ( R + ( 3 r will may be written in an equivalent form as: [( 6 ( R 6 4 ] r S 3 6R + ( Rr. (13 From Blundon s inequality S R + ( r it follows that to demonstrate the inequality (13 it will be sufficient to prove that: [( 6 ( R 6 4 ] [ ( r R + 3 ] 3 4 r 3 ( 6R Rr or ( ( 1 x x x

15 Vol. 8 A new refinement of the inequality sin A 4R+r r 9 or [( (x x ] , But who are true. ( = > 0, Corollary 1.9. In all triangle ABC holds: Proof. From the identity: Corollary 1.3 and it shall result inequality of the statement. S (4R + r ( R + ( 3 r. 3R cos A = 4R + r R, (14 cos A 1 ( A cos, 3 Corollary In all triangle ABC holds: S 6R (4R + r 3 3 R. Proof. Result from the identity (14, Jensen s inequality and Corollary 1.4 we get: cos A ( 1 3 A cos 3 4R + r R 1 ( S + 3 3R 3 16R. After performing some calculation we shall obtain the inequality of the statement. Corollary In all triangle ABC holds: Proof. Result from inequality equality (14 and (1. S 6 ( 3 (4R + r ( R + ( 1 r. R cos A 1 ( A cos, 3

16 10 Marius Dragan and Mihály Bencze No. 4 References [1] M. Bencze, Shanhe We, An refinement of the inequality sin A 3, Octogon Mathematical Magazine. [] D. S. Mitrinovic, J. E. Pekaric, V. Volonec, Recent advances in Geometric Inequalities. [3] S. Radulescu, M. Dragan, I. V. Maftei, Cateva inegalitati curadicali in Geometria Triunghiului, Revista Arhimede, Nr. 1-1/009.

17 Scientia Magna Vol. 8 (01, No. 4, On certain subclasses of analytic functions Sukhwinder Singh Billing Department of Applied Sciences, Baba Banda Singh Bahadur Engineering College Fatehgarh Sahib, , Punjab, India ssbilling@gmail.com Abstract This paper is an application of a lemma due to Miller and Mocanu [1], using which we study two certain subclasses of analytic functions and improve certain known results. Keywords Analytic function, univalent function, differential subordination. 1. Introduction and preliminaries Let H be the class of functions analytic in the open unit disk E = {z : z < 1}. Let A be the class of all functions f which are analytic in E and normalized by the conditions that f(0 = f (0 1 = 0. Thus, f A, has the Taylor series expansion f(z = z + a k z k. Let S denote the class of all analytic functions f A which are univalent in E. For two analytic functions f and g in the unit disk E, we say that f is subordinate to g in E and write as f g if there exists a Schwarz function w analytic in E with w(0 = 0 and w(z < 1, z E such that f(z = g(w(z, z E. In case the function g is univalent, the above subordination is equivalent to : f(0 = g(0 and f(e g(e. Let φ : C E C and let h be univalent in E. If p is analytic in E and satisfies the differential subordination k= φ(p(z, zp (z; z h(z, φ(p(0, 0; 0 = h(0, (1 then p is called a solution of the first order differential subordination (1. The univalent function q is called a dominant of the differential subordination (1 if p(0 = q(0 and p q for all p satisfying (1. A dominant q that satisfies q q for all dominants q of (1, is said to be the best dominant of (1. In 005, Kyohei Ochiai [] studied the classes M(α and N (α defined below: Let { M(α = f A : f(z zf (z 1 α < 1 } α, 0 < α < 1, z E, and they defined the class N (α as f(z N (α if and only if zf (z M(α. They proved the following results.

18 1 Sukhwinder Singh Billing No. 4 Theorem 1.1. If f A satisfies zf ( (z f (z 1 + zf (z f < 1 α (z for some α( 1/4 α < 1/, then f(z zf (z 1 < 1 1, z E, α therefore, f M(α. Theorem 1.. If f A satisfies zf (z f (z z(f (z + zf (z f (z + zf (z < 1 α for some α( 1/4 α < 1/, then f (z f (z + zf (z 1 < 1 1, z E, α therefore, f N (α. To prove our main result, we shall use the following lemma of Miller and Mocanu [1]. Lemma 1.1. Let q, q(z 0 be univalent in E such that zq (z is starlike in E. If an q(z analytic function p, p(z 0 in E, satisfies the differential subordination zp (z p(z zq (z q(z = h(z, then and q is the best dominant. [ z p(z q(z = exp 0 h(t t ] dt. Main results and applications E. If f A, then Theorem.1. Let q, q(z 0 be univalent in E such that zq (z (= h(z is starlike in q(z f(z zf 0 for all z in E, satisfies (z Proof. By setting p(z = zf ( (z f(z 1 + zf (z f h(z, z E, (z [ f(z z zf q(z = exp (z 0 f(z zf (z h(t t ] dt. in Lemma 1.1, proof follows.

19 Vol. 8 On certain subclasses of analytic functions 13 E. If f A, Theorem.. Let q, q(z 0 be univalent in E such that zq (z (= h(z is starlike in q(z for all z in E, satisfies then f (z f (z + zf (z 0 zf (z f (z z(f (z + zf (z f (z + zf h(z, z E, (z f [ (z z f (z + zf q(z = exp (z 0 h(t t ] dt. f (z Proof. By setting p(z = f (z + zf in Lemma 1.1, proof follows. (z Remark.1. Consider the dominant q(z = in above theorem, we have ( R 1 + zq (z q (z zq (z q(z 1 + (1 αz, 0 α < 1, z E 1 z ( = R (1 αz > 0, z E 1 z 1 + (1 αz for all 0 α < 1. Therefore, zq (z is starlike in E and we immediately get the following result. q(z f(z Theorem.3. If f A, zf 0 for all z in E, satisfies (z zf ( (z f(z 1 + zf (z (1 αz f (z (1 z(1 + (1 αz, then then f(z zf (z Theorem.4. Let f A, 1 + (1 αz, 0 α < 1, z E. 1 z f (z f (z + zf 0 for all z in E, satisfy (z zf (z f (z z(f (z + zf (z (1 αz f (z + zf (z (1 z(1 + (1 αz, f (z f (z + zf (z 1 + (1 αz, 0 α < 1, z E. 1 z Note that Theorem.3 is more general than the result of Kyohei Ochiai [] stated in Theorem 1.1 and similarly Theorem.4 is the general form of Theorem 1.. ( δ 1 + z Remark.. Selecting the dominant q(z =, 0 < δ 1, z E, we have 1 z ( ( R 1 + zq (z q (z zq (z 1 + z = R q(z 1 z > 0, z E.

20 14 Sukhwinder Singh Billing No. 4 zq (z Therefore, is starlike in E and from Theorem.1 and Theorem., we obtain the q(z following results, respectively. f(z Theorem.5. Suppose that f A, zf 0 for all z in E, satisfies (z zf ( (z f(z 1 + zf (z f δz (z 1 z, then then Theorem.6. Let f A, f(z zf (z ( δ 1 + z, 0 < δ 1, z E. 1 z f (z f (z + zf 0 for all z in E, satisfy (z zf (z f (z z(f (z + zf (z f (z + zf δz (z 1 z, f (z f (z + zf (z ( δ 1 + z, 0 < δ 1, z E. 1 z Remark.3. When we select the dominant q(z =.1 and Theorem.. A little calculation yields ( R 1 + zq (z q (z zq (z = R q(z Therefore, zq (z q(z ( 1 1 z + z α z is starlike in E and we get the following results. α(1 z, α > 1, z E in Theorem α z > 0, z E. Theorem.7. Suppose that α > 1 is a real number and if f A, E, satisfies then zf ( (z f(z 1 + zf (z (1 αz f (z (1 z(α z, f(z α(1 z zf (z α z, z E. Theorem.8. Let α > 1 be a real number and let f A, E, satisfy then zf (z f (z z(f (z + zf (z (1 αz f (z + zf (z (1 z(α z, f (z α(1 z f (z + zf (z α z, z E. f(z zf 0 for all z in (z f (z f (z + zf 0 for all z in (z Remark.4. Consider the dominant q(z = 1 + λz, 0 < λ 1, z E in Theorem.1 and Theorem., we have ( ( R 1 + zq (z q (z zq (z 1 = R > 0, z E q(z 1 + λz

21 Vol. 8 On certain subclasses of analytic functions 15 for all 0 < λ 1. Therefore, zq (z q(z then Theorem.9. Suppose f A, zf (z f(z Theorem.10. Suppose f A, is starlike in E and we have the following results. f(z zf 0 for all z in E, satisfies (z ( 1 + zf (z f λz (z 1 + λz, f(z zf (z 1 < λ, 0 < λ 1, z E. f (z f (z + zf 0 for all z in E, satisfies (z zf (z f (z z(f (z + zf (z f (z + zf λz (z 1 + λz, then f (z f (z + zf (z 1 < λ, 0 < λ 1, z E. Remark.5. We, now claim that Theorem.9 extends Theorem 1.1 in the sense that the operator zf ( (z 1 + zf (z f(z f, now, takes values in an extended and Theorem.10 gives (z the same extension for Theorem 1.. We, now, compare the results by taking the following particular cases. Setting λ = 1 in Theorem.9, we obtain: f(z Suppose f A, zf 0 for all z in E, satisfies (z then zf ( (z f(z 1 + zf (z f (z f(z zf (z 1 < 1, z E. z 1 + z, ( For α = 1/4, Theorem 1.1 reduces to the following result: If f A satisfies zf ( (z f (z 1 + zf (z f < 1 (z, (3 then f(z zf (z 1 < 1, z E, According to the result stated in (3, the operator zf ( (z 1 + zf (z f(z f takes values (z within the disk of radius 1/ and centered at origin (as shown by the dark shaded portion in Figure.1 to give the conclusion that f(z zf (z 1 < 1, whereas in view of the result stated above in (, the same operator can take values in the entire shaded region (dark + light in Figure.1 to get the same conclusion. Thus the result stated in ( extends the result stated above in (3 in the sense that the region in which the operator zf ( (z f(z 1 + zf (z f takes (z values is extended. The claimed extension is given by the light shaded portion of Figure 1. In the same fashion, the above explained extension also holds in comparison of results in Theorem.10 and Theorem 1..

22 16 Sukhwinder Singh Billing No. 4 References Figure 1 [1] S. S. Miller and P. T. Mocanu, Differential Subordinations : Theory and Applications, Marcel Dekker, New York and Basel, No. 5, 000. [] K. Ochiai, S. Owa and M. Acu, Applications of Jack s lemma for certain subclasses of analytic functions, General Mathematics, 13(005, No., 73-8.

23 Scientia Magna Vol. 8 (01, No. 4, 17-4 Certain indefinite integral associate to elliptic integral and complex argument Salahuddin P. D. M College of Engineering, Bahadurgarh, Haryana, India vsludn@gmail.com sludn@yahoo.com Abstract In this paper we have developed some indefinite integrals. The results represent here are assume to be new. Keywords Floor function, elliptic integral. 1. Introduction and preliminaries Definition 1.1. Floor and ceiling functions. In mathematics and computer science, the floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively. More precisely, f loor(x = x is the largest integer not greater than x and ceiling(x = x is the smallest integer not less than x. The floor function is also called the greatest integer or entier (French for integer function, and its value at x is called the integral part or integer part of x. In the following formulas, x and y are real numbers, k, m, and n are integers, and Z is the set of integers (positive, negative, and zero. Floor and ceiling may be defined by the set equations x = max{m Z m x}, x = min{n Z n x}. Since there is exactly one integer in a half-open interval of length one, for any real x there are unique integers m and n satisfying x 1 < m x n < x + 1. Then x = m and x = n may also be taken as the definition of floor and ceiling. These formulas can be used to simplify expressions involving floors and ceilings. x = m, if and only if, m x < m + 1. x = n, if and only if, n 1 < x n. x = m, if and only if, x 1 < m x. x = n, if and only if, x n < x + 1. In the language of order theory, the floor function is a residuated mapping, that is, part of a Galois connection: it is the upper adjoint of the function that embeds the integers into the reals. x < n, if and only if, x < n. n < x, if and only if, n < x. x n, if and only if, x n. n x, if and only if, n x.

24 18 Salahuddin No. 4 These formulas show how adding integers to the arguments affect the functions: x + n = x + n, x + n = x + n. The above are not necessarily true if n is not an integer, however: x + y x + y x + y + 1, x + y 1 x + y x + y. Definition 1.. Elliptic integral. In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler. Modern mathematics defines an elliptic integral as any function f which can be expressed in the form f(x = x (t, c R P (t dt, where R is a rational function of its two arguments, P a polynomial of degree 3 or 4 with no repeated roots, and c is a constant. In general, elliptic integrals cannot be expressed in terms of elementary functions. Exceptions to this general rule are when P has repeated roots, or when R(x, y contains no odd powers of y. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms (i.e. the elliptic integrals of the first, second and third kind. Besides the Legendre form, the elliptic integrals may also be expressed in Carlson symmetric form. Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz-Christoffel mapping. Historically, elliptic functions were discovered as inverse functions of elliptic integrals. Incomplete elliptic integrals are functions of two arguments, complete elliptic integrals are functions of a single argument. Definition 1.3. The incomplete elliptic integral of the first kind F is defined as F (ψ, k = F (ψ k = F (sin ψ; k = ψ 0 dθ 1 k sin θ. This is the trigonometric form of the integral, substituting t = sin θ, x = sin ψ, one obtains Jacobi s form x dt F (x; k = (1 t (1 k t. Equivalently, in terms of the amplitude and modular angle one has: F (ψ\α = F (ψ, sin α = 0 ψ 0 dθ 1 (sin θ sin α. In this notation, the use of a vertical bar as delimiter indicates that the argument following it is the parameter (as defined above, while the backslash indicates that it is the modular angle. The use of a semicolon implies that the argument preceding it is the sine of the amplitude: Definition 1.4. F (ψ, sin α = F (ψ sin α = F (ψ\α = F (sin ψ; sin α. Incomplete elliptic integral of the second kind E is defined as E(ψ, k = E(ψ k = E(sin ψ; k = 1 k sin θ dθ.

25 Vol. 8 Certain indefinite integral associate to elliptic integral and complex argument 19 or Substituting t = sin θ and x = sin ψ, one obtains Jacobi s form: x 1 k t E(x; k = dt. 1 t Equivalently, in terms of the amplitude and modular angle: Definition 1.5. E(ψ\α = E(ψ, sin α = 0 ψ 0 1 (sin θ sin α dθ. Incomplete elliptic integral of the third kind Π is defined as Π(n; ψ\α = Π(n; ψ m = ψ 0 sin ψ n sin θ dθ 1 (sin θ sin α 1 dt 1 nt (1 mt (1 t. The number n is called the characteristic and can take on any value, independently of the other arguments. Definition 1.6. Incomplete elliptic integral of the third kind Π is defined as or Π(n; ψ\α = Π(n; ψ m = ψ 0 sin ψ n sin θ dθ 1 (sin θ sin α 1 dt 1 nt (1 mt (1 t. The number n is called the characteristic and can take on any value, independently of the other arguments. Definition 1.7. Complete elliptic integral of the first kind is defined as Elliptic Integrals are said to be complete when the amplitude ψ = π and therefore x=1. The complete elliptic integral of the first kind K may thus be defined as K(k = π 0 dθ 1 1 k sin θ = 0 dt (1 t (1 k t or more compactly in terms of the incomplete integral of the first kind as ( π K(k = F, k = F (1; k. It can be expressed as a power series K(k = π [ ] (n! n (n! k n = π [ k P n (0] n, n=0 where P n is the Legendre polynomial, which is equivalent to [ K(k = π ( ( { } (n 1!! 1 + k + k k n +...],. 4 (n!! n=0

26 0 Salahuddin No. 4 where n!! denotes the double factorial. In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as K(k = π F 1 ( 1, 1 ; 1; k. The complete elliptic integral of the first kind is sometimes called the quarter period. It can most efficiently be computed in terms of the arithmetic-geometric mean K(k = π agm(1 k, 1 + k. Definition 1.8. Complete elliptic integral of the second kind is defined as The complete elliptic integral of the second kind E is proportional to the circumference of the ellipse C C = 4aE(e, where a is the semi-major axis, and e is the eccentricity. E may be defined as π 1 1 E(k = 1 k sin k t θ dθ = dt t or more compactly in terms of the incomplete integral of the second kind as ( π E(k = E, k = E(1; k. It can be expressed as a power series which is equivalent to [ E(k = π 1 ( 1 k 1 E(k = π n=0 [ ] (n! k n n (n! 1 n, ( 1. 3 k { } ] (n 1!! k n (n!! n In terms of the Gauss hypergeometric function, the complete elliptic integral of the second kind can be expressed as E(k = π F 1 ( 1, 1 ; 1; k. Definition 1.9. Complete elliptic integral of the third kind is defined as The complete elliptic integral of the third kind Π can be defined as Π(n, k = π 0 dθ (1 n sin θ 1 k sin θ. Argument. In mathematics, arg is a function operating on complex numbers (visualised as a flat plane. It gives the angle between the line joining the point to the origin and the positive real axis, shown as ψ in figure 1, known as an argument of the point (that is, the angle between the half-lines of the position vector representing the number and the positive real axis.

27 Vol. 8 Certain indefinite integral associate to elliptic integral and complex argument 1 Figure 1. In figure 1, The Argand diagram represents the complex numbers lying on a plane. For each point on the plane, arg is the function which returns the angle ψ. A complex number may be represented as z = x + ιy = z e ιθ where z is a positive real number called the complex modulus of z, and ψ is a real number called the argument. The argument is sometimes also known as the phase or, more rarely and more confusingly, the amplitude [7]. The complex argument can be computed as ( arg(x + ιy tan 1 y. x. Main integrals ι = dy (1 x cosh y ( F ιy 1 x cosh y x cosh y 1 x 1 ι x cosh y 1F = x x 1 + C ( ( ιy x x 1 exp ιπ arg(x 1 arg(x cosh y 1 π π C. (1 x 1 1 x cosh y

28 Salahuddin No. 4 = = dy (1 x sinh y ( x sinh y 1 1 ι 1 ιx F 4 (π 4ιy x x ι + C 1 x sinh y ι ( ( 1 x sinh y 1F 4 (π 4ιy x x ι exp ιπ π ιx 1 1 x sinh y arg( ιx 1 arg(x sinh y 1 π C.( = dy (1 x tanh y ( [ x tanh 1 tanh y 1 x 1 x 1 tanh 1 1 x tanh y ( x tanh y 1 ] x 1 x x 1 tanh y 1 + C. (3 = dy (1 x coth y ( [ ι(x tanh 1 coth y 1 ι(x 1 ι(x 1 tanh 1 ( ι(x coth y 1 ] ι(x+1 ι(x coth y 1 ι(x+1 1 x coth y ι ι(x 1 ( ι(x coth y 1 tanh 1 ι(x coth y 1 ι(x 1 = (x 1 1 x coth y ι ι(x + 1 ( ι(x coth y 1 tanh 1 ι(x coth y 1 ι(x+1 (x C. (4 1 x coth y + C 1 x sinh ydy ( x sinh y 1 1 (x ι 1 ιx E 4 (π 4ιy x x ι = + Constant 1 x sinh y ι ( ( 1 x sinh y 1E 4 (π 4ιy x x ι exp ιπ arg( ιx 1 π = ιx 1 1 x sinh y x ( ( 1 x sinh y 1E 4 (π 4ιy x x ι exp ιπ arg( ιx 1 π ιx 1 1 x sinh y arg(x sinh y 1 π + 1 arg(x sinh y 1 π + 1 +C. (5

29 Vol. 8 Certain indefinite integral associate to elliptic integral and complex argument 3 1 x cosh ydy = = ι(x ι x cosh y 1 x 1 ( E ιy x x 1 1 x cosh y 1 ι x 1 x cosh y 1 1 x cosh y + Constant ( x ( arg(x 1 arg(x cosh y 1 E ιy exp ιπ C. (6 x 1 π π 1 x tanh ydy = [{ ( 1 x 1 tanh 1 x tanh y 1 x tanh y 1 x 1 x 1 tanh 1 ( x tanh y 1 x 1 } 1 x tanh y 1 ] + C. (7 1 xcosech ydy = 1 ι cosech ysech y xcosech ysech y [ { ι(xcosechy 1 sech y x 1 + ι cosech y(1 + ι sinh y cosh y ( ( F sin sinh y ( Π ; sin 1 ( 1 x ι x ιcosech ysech y x ι coth y 1 ιcosech y ιcosech ysech y x x ι }] + C. (8 1 x coth ydy = 1 ι(x coth y 1 [{ ( ι(x 1 tanh 1 ι(x coth y 1 ι(x 1 ι(x + 1 tanh 1 ( ι(x coth y 1 ι(x + 1 } 1 x coth y ] + C. (9 where C denotes constant. Derivations. By involving the method of [18] one can derived the integrals. References [1] Milton Abramowitz, Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, 1965.

30 4 Salahuddin No. 4 [] Lars V. Ahlfors, Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, New York, London, [3] Alfred George Greenhill, The applications of elliptic functions, New York, Macmillan. [4] A. F. Beardon, Complex Analysis: The Argument Principle in Analysis and Topology, Chichester, [5] Borowski Ephraim, Borwein Jonathan, Mathematics, Collins Dictionary (nd ed.. Glasgow, 00. [6] Crandall Richard, Pomeramce Carl, Prime Numbers: A Computational Perspective, New York, 001. [7] Derbyshire John.., Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, New York, 004. [8] L. Graham Ronald, E. Knuth Donald, Patashnik Oren, Concrete Mathematics, Reading Ma., [9] G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, Oxford, [10] Harris Hancock, Lectures on the theory of Elliptic functions, New York, J. Wiley & sons, [11] J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, 45(1957. [1] Steven G. Krantz, Handbook of Complex Variables, Boston, MA, [13] Lemmermeyer Franz, Reciprocity Laws: from Euler to Eisenstein, Berlin, 000. [14] Louis V. King, On The Direct Numerical Calculation Of Elliptic Functions And Integrals, Cambridge University Press, 194. [15] Press WH, Teukolsky SA, Vetterling WT, Flannery BP, Numerical Recipes: The Art of Scientific Computing, New York, 007. [16] Ramanujan Srinivasa, Collected Papers, Providence RI: AMS/Chelsea, 000. [17] Ribenboim Paulo, The New Book of Prime Number Records, New York, [18] Salahuddin., Hypergeometric Form of Certain Indefinite Integrals, Global Journal of Science Frontier Research, 1(01, [18] R. A. Silverman, Introductory Complex Analysis, New York [18] Titchmarsh Edward Charles, Heath-Brown David Rodney, The Theory of the Riemann Zeta function, Oxford, 1986.

31 Scientia Magna Vol. 8 (01, No. 4, 5-6 On the types and sums of self-intergrating functions A. C. F. Bueno Department of Mathematics and Physics, Central Luzon State University, Science City of Muñoz 310, Nueva Ecija, Philippines Abstract In this paper, two types of self-inetgrability of functions and the underlying conditions for each type was presented. Furthermore, the sum of self-integrable functions was investigated. Keywords self-integrability, absolutely self-integrating, conditionally self-integrating. 1. Introduction and preliminaries Graham [1] shown that the polynomials of the form p k (x = x k + has the property that 1 0 p k(x dx = p k (1 p k (0. The said polynomial is self-integrating in the closed interval [0,1]. if Definition 1.1. A function f is said to be self-integrating on an interval [a,b] if and only b a f(x dx = f(b f(a. The following are the types of self-integrability: k k+1 Definition 1.. Absolute self-integrability is the self-integrability everywhere in R. Definition 1.3. Conditional self-integrability is the self-integrability in some interval [a,b]. Definition 1.4. A function f is said to be absolutely self-integrating if and only if b a f(x dx = f(b f(a for every [a,b]. Definition 1.5. A function f is said to be conditionally self-integrating if and only if b a f(x dx = f(b f(a for some [a,b]. The natural exponential function and the zero function are absolutely self-integrating while any polynomial of the form p k (x = x k + is conditionally self-integrating. k k+1

32 6 A. C. F. Bueno No. 4. Main results Theorem.1. The sum of absolutely self-integrating functions is also absolutely (conditionally self-integrating. Proof. Suppose f 0 (x, f 1 (x,..., f n (x are absolutely self-integrating functions. Taking their sum and the definite integral for any closed interval [a,b] yields and b a n 1 k=0 n 1 k=0 f k (x f k (x dx. Since each function is absolutely self-integrating, References b a n 1 k=0 f k (x dx = n 1 k=0 n 1 f k (b k=0 f k (a. [1] J. Graham, Self-Integrating Polynomials, The College Mathematics Journal, 36(005 No.4,

33 Scientia Magna Vol. 8 (01, No. 4, 7-4 Creation of a summation formula enmeshed with contiguous relation Salah Uddin Abstract P. D. M College of Engineering, Bahadurgarh, Haryana, India sludn@yahoo.com vsludn@gmail.com The main aim of the present paper is to create a summation formula enmeshed with contiguous relation and recurrence relation. Keywords Gaussian hypergeometric function, contiguous function, recurrence relation, Bai -ley summation theorem and Legendre duplication formula. 010 Mathematics Subject Classification: 33C60, 33C70, 33D15, 33D50, 33D Introduction Generalized Gaussian hypergeometric function of one variable is defined by AF B a 1, a,, a A ; z (a 1 k (a k (a A k z k = b 1, b,, b B ; (b 1 k (b k (b B k k! k=0 or AF B (a A; (b B ; z A F B (a j A j=1 ; (b j B j=1 ; z = k=0 ((a A k z k ((b B k k!, (1 where the parameters b 1, b,, b B are neither zero nor negative integers and A, B are nonnegative integers. Definition 1.1. Contiguous relation [1] is defined as follows a, b; a + 1, b; a, b + 1; (a b F 1 z = a F 1 z b F 1 z. ( c; c; c; Definition 1.. Recurrence relation of gamma function is defined as follows Γ(z + 1 = zγ(z. (3 Definition 1.3. Legendre duplication formula [3] is defined as follows ( πγ(z = (z 1 Γ(z Γ z + 1. (4

34 8 Salah Uddin No. 4 ( 1 Γ = π = (b 1 Γ( b b+1 Γ( Γ(b ( 1 Γ = (a 1 Γ( a a+1 Γ( Γ(a. (5. (6 Definition 1.4. Bailey summation theorem [4] is defined as follows a, 1 a; F 1 1 = Γ( c c+1 Γ( πγ(c c; Γ( c+a c+1 a Γ( = c 1 Γ( c+a c+1 a Γ(. (7. Main results of summation formula a, a 47; F 1 1 πγ(c = c; c a Γ( a Γ( a3 Γ( a4 Γ( [ Γ( a a 6 Γ( a a 8 Γ( a a 10 Γ( a a 1 Γ( a a a 15 Γ( a a a a 19 Γ( a a a c Γ( ac Γ( a c Γ(

35 Vol. 8 Creation of a summation formula enmeshed with contiguous relation a3 c Γ( a4 c a 5 c Γ( a6 c a 7 c Γ( a8 c a 9 c Γ( a10 c a 11 c Γ( a1 c a 13 c Γ( a14 c a 15 c a 16 c a17 c a 18 c Γ( a19 c a 0 c 4816a 1 c Γ( Γ( + 48a c c Γ( ac Γ( a c Γ( a3 c Γ( a4 c Γ( a5 c Γ( a6 c a 7 c Γ( a8 c a 9 c Γ( a10 c a 11 c Γ( a1 c a 13 c Γ( a14 c a 15 c a 16 c Γ( a17 c a 18 c a 19 c a 0 c Γ( c3 Γ(

36 30 Salah Uddin No ac3 Γ( a c 3 Γ( a3 c 3 Γ( a4 c 3 Γ( a5 c a 6 c 3 Γ( a7 c a 8 c 3 Γ( a9 c a 10 c 3 Γ( a11 c a 1 c 3 Γ( a13 c a 14 c a 15 c 3 Γ( a16 c a 17 c a 18 c a 19 c 3 Γ( a0 c c 4 Γ( ac4 Γ( a c 4 Γ( a3 c 4 Γ( a4 c 4 Γ( a5 c 4 Γ( a6 c a 7 c 4 Γ( a8 c a 9 c 4 Γ( a10 c a 11 c 4 Γ( a1 c a 13 c a 14 c 4 Γ( a15 c a 16 c a 17 c a 18 c c5 Γ( Γ(

37 Vol. 8 Creation of a summation formula enmeshed with contiguous relation ac5 Γ( a c 5 Γ( a3 c 5 Γ( a4 c a 5 c 5 Γ( a6 c a 7 c 5 Γ( a8 c a 9 c 5 Γ( a10 c a 11 c 5 Γ( a1 c a 13 c a 14 c 5 Γ( a15 c a 16 c a 17 c a 18 c 5 Γ( c6 Γ( ac6 Γ( a c 6 Γ( a3 c 6 Γ( a4 c 6 Γ( a5 c 6 Γ( a6 c a 7 c 6 Γ( a8 c a 9 c 6 Γ( a10 c a 11 c 6 Γ( a1 c a 13 c 6 Γ( a14 c a 15 c a 16 c 6 Γ( c ac 7 Γ(

38 3 Salah Uddin No a c a 3 c 7 Γ( a4 c a 5 c 7 Γ( a6 c a 7 c 7 Γ( a8 c a 9 c 7 Γ( a10 c a 11 c a 1 c 7 Γ( a13 c a 14 c a 15 c a 16 c 7 Γ( c ac 8 Γ( a c a 3 c 8 Γ( a4 c a 5 c 8 Γ( a6 c a 7 c 8 Γ( a8 c a 9 c 8 Γ( a10 c a 11 c a 1 c 8 Γ( a13 c a 1 4c c 9 Γ( ac a c 9 Γ( a3 c a 4 c 9 Γ( a5 c a 6 c 9 Γ( a7 c a 8 c 9 Γ( a9 c a 10 c 9 Γ( a11 c a 1 c 9 Γ( a13 c a 14 c c 10 Γ( ac a c 10 Γ(

39 Vol. 8 Creation of a summation formula enmeshed with contiguous relation a3 c a 4 c 10 Γ( a5 c a 6 c 10 Γ( a7 c a 8 c a 9 c 10 Γ( a10 c a 11 c a 1 c 10 Γ( c ac 11 Γ( a c a 3 c 11 Γ( a4 c a 5 c 11 Γ( a6 c a 7 c 11 Γ( a8 c a 9 c 11 Γ( a1 0c a 11 c a 1 c 11 Γ( c ac 1 Γ( a c a 3 c 1 Γ( a4 c a 5 c 1 Γ( a6 c a 7 c a 8 c 1 Γ( a9 c a 10 c c 13 Γ( ac a c 13 Γ( a3 c a 4 c 13 Γ( a5 c a 6 c a 7 c 13 Γ( a8 c a 9 c a 10 c 13 Γ( c ac 14 Γ(

40 34 Salah Uddin No a c a 3 c 14 Γ( a4 c a 5 c a 6 c 14 Γ( a7 c a 8 c c 15 Γ( ac a c 15 Γ( a3 c a 4 c a 5 c 15 Γ( a6 c a 7 c a 8 c 15 Γ( c ac 16 Γ( a c a 3 c 16 Γ( a4 c a 5 c 16 Γ( a6 c c ac 17 Γ( a c a 3 c a 4 c 17 Γ( a5 c a 6 c c 18 Γ( ac a c a 3 c 18 Γ( a4 c c ac 19 Γ( a c a 3 c a 4 c c 0 Γ( ac a c c ac 1 Γ( a c c c 3 Γ( a Γ( c+a a Γ( c+a a3 Γ( c+a a4 Γ( c+a+48

41 Vol. 8 Creation of a summation formula enmeshed with contiguous relation a a 6 Γ( c+a a a 8 Γ( c+a a a 10 Γ( c+a a a 1 Γ( c+a a a 14 Γ( c+a a a a 17 Γ( c+a a a a a 1 090a Γ( c+a a3 + a c Γ( c+a ac Γ( c+a a c Γ( c+a a3 c Γ( c+a a4 c Γ( c+a a5 c Γ( c+a a6 c Γ( c+a a7 c a 8 c Γ( c+a a9 c a 10 c Γ( c+a a11 c a 1 c Γ( c+a a13 c a 14 c Γ( c+a a15 c a 16 c a 17 c Γ( c+a a18 c a 19 c a 0 c a 1 c 1348a c Γ( c+a c Γ( c+a ac Γ( c+a a c Γ( c+a+48

42 36 Salah Uddin No a3 c Γ( c+a a4 c Γ( c+a a5 c Γ( c+a a6 c Γ( c+a a7 c a 8 c Γ( c+a a9 c a 10 c Γ( c+a a11 c a 1 c Γ( c+a a13 c a 14 c Γ( c+a a15 c a 16 c a 17 c Γ( c+a a18 c a 19 c a 0 c a 1 c 88a c Γ( c+a c3 Γ( c+a ac3 Γ( c+a a c 3 Γ( c+a a3 c 3 Γ( c+a a4 c 3 Γ( c+a a5 c 3 Γ( c+a a6 c a 7 c 3 Γ( c+a a8 c a 9 c 3 Γ( c+a a10 c a 11 c 3 Γ( c+a a1 c a 13 c 3 Γ( c+a a14 c a 15 c a 16 c 3 Γ( c+a a17 c a 18 c a 19 c a 0 c 3 Γ( c+a c4 Γ( c+a+48

43 Vol. 8 Creation of a summation formula enmeshed with contiguous relation ac4 Γ( c+a a c 4 Γ( c+a a3 c 4 Γ( c+a a4 c 4 Γ( c+a a5 c 4 Γ( c+a a6 c a 7 c 4 Γ( c+a a8 c a 9 c 4 Γ( c+a a10 c a 11 c 4 Γ( c+a a1 c a 13 c 4 Γ( c+a a14 c a 15 c a 16 c 4 Γ( c+a a17 c a 18 c a 19 c a 0 c 4 Γ( c+a c5 Γ( c+a ac5 Γ( c+a a c 5 Γ( c+a a3 c 5 Γ( c+a a4 c 5 Γ( c+a a5 c a 6 c 5 Γ( c+a a7 c a 8 c 5 Γ( c+a a9 c a 10 c 5 Γ( c+a a11 c a 1 c 5 Γ( c+a a13 c a 14 c a 15 c 5 Γ( c+a a16 c a 17 c a 18 c 5 Γ( c+a+48

44 38 Salah Uddin No c6 Γ( c+a ac6 Γ( c+a a c 6 Γ( c+a a3 c 6 Γ( c+a a4 c a 5 c 6 Γ( c+a a6 c a 7 c 6 Γ( c+a a8 c a 9 c 6 Γ( c+a a10 c a 11 c 6 Γ( c+a a1 c a 13 c a 14 c 6 Γ( c+a a15 c a 16 c a 17 c a 18 c 6 Γ( c+a c7 Γ( c+a ac7 Γ( c+a a c 7 Γ( c+a a3 c 7 Γ( c+a a4 c a 5 c 7 Γ( c+a a6 c a 7 c 7 Γ( c+a a8 c a 9 c 7 Γ( c+a a10 c a 11 c a 1 c 7 Γ( c+a a13 c a 14 c a 15 c 7 Γ( c+a a16 c c 8 Γ( c+a ac8 Γ( c+a a c a 3 c 8 Γ( c+a+48

45 Vol. 8 Creation of a summation formula enmeshed with contiguous relation a4 c a 5 c 8 Γ( c+a a6 c a 7 c 8 Γ( c+a a8 c a 9 c 8 Γ( c+a a10 c a 11 c a 1 c 8 Γ( c+a a13 c a 14 c a 15 c a 16 c 8 Γ( c+a c ac 9 Γ( c+a a c a 3 c 9 Γ( c+a a4 c a 5 c 9 Γ( c+a a6 c a 7 c 9 Γ( c+a a8 c a 9 c 9 Γ( c+a a10 c a 11 c a 1 c 9 Γ( c+a a13 c a 14 c c 10 Γ( c+a ac a c 10 Γ( c+a a3 c a 4 c 10 Γ( c+a a5 c a 6 c 10 Γ( c+a a7 c a 8 c 10 Γ( c+a a9 c a 10 c a 11 c 10 Γ( c+a a1 c a 13 c a 14 c 10 Γ( c+a c ac 11 Γ( c+a a c a 3 c 11 Γ( c+a a4 c a 5 c 11 Γ( c+a+48

46 40 Salah Uddin No a6 c a 7 c 11 Γ( c+a a8 c a 9 c a 10 c 11 Γ( c+a a11 c a 1 c c 1 Γ( c+a ac a c 1 Γ( c+a a3 c a 4 c 1 Γ( c+a a5 c a 6 c 1 Γ( c+a a7 c a 8 c a 9 c 1 Γ( c+a a10 c a 11 c a 1 c 1 Γ( c+a c ac 13 Γ( c+a a c a 3 c 13 Γ( c+a a4 c a 5 c 13 Γ( c+a a6 c a 7 c a 8 c 13 Γ( c+a a9 c a 10 c c 14 Γ( c+a ac a c 14 Γ( c+a a3 c a 4 c 14 Γ( c+a a5 c a 6 c a 7 c 14 Γ( c+a a8 c a 9 c a 10 c 14 Γ( c+a c ac 15 Γ( c+a a c a 3 c 15 Γ( c+a a4 c a 5 c a 6 c 15 Γ( c+a a7 c a 8 c c 16 Γ( c+a ac a c 16 Γ( c+a+48

47 Vol. 8 Creation of a summation formula enmeshed with contiguous relation a3 c a 4 c a 5 c 16 Γ( c+a a6 c a 7 c a 8 c 16 Γ( c+a c ac 17 Γ( c+a a c a 3 c 17 Γ( c+a a4 c a 5 c 17 Γ( c+a a6 c c ac 18 Γ( c+a a c a 3 c a 4 c 18 Γ( c+a a5 c a 6 c c 19 Γ( c+a ac a c a 3 c 19 Γ( c+a a4 c c ac 0 Γ( c+a a c a 3 c a 4 c c 1 Γ( c+a ac a c c ac Γ( c+a+48 ] a c c c 4 Γ( c+a+48. (8 Derivation of result (8: Putting in established result (, we get b = a 47, z = 1 (a + 47 F 1 a, a 47; c; 1 a + 1, a 47; = a F 1 1 c; a, a 46; +(a + 47 F 1 c; 1. Now proceeding same parallel method which is applied in [6], we can prove the main formula.

48 4 Salah Uddin No. 4 References [1] L. C. Andrews (199, Special Function of mathematics for Engineers,second Edition, McGraw-Hill Co Inc., New York. [] Arora, Asish, Singh, Rahul, Salahuddin, Development of a family of summation formulae of half argument using Gauss and Bailey theorems, Journal of Rajasthan Academy of Physical Sciences., 7(008, [3] Bells, Richard, Wong, Roderick, Special Functions, A Graduate Text. Cambridge Studies in Advanced Mathematics, 010. [4] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series Vol. 3: More Special Functions., Nauka, Moscow, Translated from the Russian by G. G. Gould, Gordon and Breach Science Publishers, New York, Philadelphia, London, Paris, Montreux, Tokyo, Melbourne, [5] E. D. Rainville, The contiguous function relations for p F q with applications to Bateman s J u,v n and Rice s H n (ζ, p, ν, Bull. Amer. Math. Soc., 51(1945, [6] Salahuddin, M. P. Chaudhary, A New Summation Formula Allied With Hypergeometric Function, Global Journal of Science Frontier Research, 11(010, [7] Salahuddin, Evaluation of a Summation Formula Involving Recurrence Relation, Gen. Math. Notes., (010, 4-59.

49 Scientia Magna Vol. 8 (01, No. 4, Distar decompositions of a symmetric product digraph V. Maheswari and A. Nagarajan Department of Mathematics, A. P. C. Mahalaxmi College, Tuticorin, Tamil Nadu, India Department of Mathematics, V. O. C. College, Tuticorin, Tamil Nadu, India mahi raj005@yahoo.com nagarajan.voc@yahoo.co.in Abstract In this paper, we decompose the symmetric crown, C n,n 1 into distars, some sufficient conditions are also given. Furthermore, we consider the problem of decomposing the (P m K n into distar S k,l -decomposition. 1. Introduction A distar S k,l is the digraph obtained from the star S k+l by directing k of the edges out of the center and l of the edges into the center. A multiple star is a star with multiple edges allowed. A directed multiple star is a digraph whose underlying graph is a multiple star recall that for a digraph H and x V (H, d(x = d + (x + d (x. The problem of isomorphic S k,l -decomposition of symmetric complete digraph Kn was posed by Caetano and Heinrich [1], and was solved by Colbourn in [3]. The problem of decomposing the symmetric complete bipartite digraph Km,n into S k,l was solved by Hung- Chih Lee in [5]. For positive integers k n, the crown C n,k is the graph with vertex set {a 1, a,..., a n, b 1, b,..., b n } and the edge set {a i b j : 1 i n, j i + 1, i +,..., i + k ( mod n}. For a graph G, G denotes the symmetric digraph of G, that is, the digraph obtained from G by replacing each of its edge by a symmetric pair of arcs. Definitions which are not seen here can be found in []. Throughout this paper, we assume that k, l 1 and n 1 are integers, and let (X, Y be the bipartition of Cn,n 1 where X = {x 1, x,..., x n } and Y = {y 1, y,..., y n }. The weak product of the graphs (resp. digraphs G and H, denoted by G H, has vertex set V (G V (H in which (g 1, h 1 (g, h is an edge (resp. arc whenever (g 1, g is an edge (resp. arc in G and (h 1, h is an edge (resp. arc in H. We use the following results in the proof of the main theorem. Lemma 1.1. Let k, l, t be positive integers. Suppose H is a directed multiple star with center w of in degree lt and out degree kt. Then H has an S k,l -decomposition if and only if e(w, x t for every end vertex x of H where e(w, x denotes the number of arcs joining w and x.

50 44 V. Maheswari and A. Nagarajan No. 4 Lemma 1.. {((p 1, q 1, (p, q,..., (p m, q m, ((s 1, t 1, (s, t,..., (s n, t n } is bipartite digraphical if and only if {(p 1, p,..., p m, (t 1, t,..., t n } and {(q 1, q,..., q m, (s 1, s,..., s n } are both bigraphical. Theorem 1.1. The pair {(a 1, a,..., a m, (b 1, b,..., b n } of the non-negative integral sequences with a 1 a... a m is bigraphical if and only if r i=1 a i n j=1 min{b j, r} for 1 r m and m i=1 a i = n j=1 b j. Definition 1.1. Let (a 1, a,..., a m, (b 1, b,..., b n be two sequences of non-negative integers. The pair {(a 1, a,..., a m, (b 1, b,..., b n } is called bigraphical if there exists a bipartite graph G with bipartition (M, N, where M = {x 1, x,..., x m } and N = {y 1, y,..., y n } such that for i = 1,,..., m and j = 1,,..., n. d G (x i = a i and d G (y j = b j. A simple digraph is a digraph in which each ordered pair of vertices occurs as an arc at most once. Let ((p 1, q 1, (p, q,..., (p m, q m, ((s 1, t 1, (s, t,..., (s n, t n be two sequences of non-negative integral ordered pair. The pair {((p 1, q 1, (p, q,..., (p m, q m, ((s 1, t 1, (s, t,..., (s n, t n } is bipartite digraphical if there exists a simple bipartite digraph H with bipartition (M, N, where M = {x 1, x,..., x m } and N = {y 1, y,..., y n } such that for i = 1,,..., m and j = 1,,..., n. d H (x i = p i, d + H (x i = q i, d H (y j = s j, d + H (y j = t j.. Distar decompositions Theorem.1. Suppose C n,n 1 has an S k,l -decomposition. Then (i If k + l n 1, then k = l and k n 1. (ii If k + l n 1, k = l, then k n 1. (iii If k + l n 1, k l, then (k + l n(n 1 and n 1 k+l min{ n 1 k, n 1 }. (iv If k + l = n 1, k l, then (k + l n(n 1. Proof. Let D be an S k,l -decomposition of C n,n 1. For i = 1,,..., n, let p i and q j be the number of S k,l in D with centers at x i and y j, respectively. First prove (i. Suppose k + l n 1, then any distar S k,l in D may have its center in M or in N. Hence n 1 = d (x i = lp i, n 1 = d + (x i = kp i. Thus k = l and k n 1. This completes (i. Consider (ii. Suppose k + l n 1. Let G be a sub digraph of C n,n 1 such that the arcs of G are those of S k,l in D which are oriented to the centers. Then d G (x i = lp i, d + G (x i = n 1 kp i, d G (y j = lq j d + G (y j = n 1 kq j. Thus Hence n lp i = i=1 n lq j = j=1 n d G (x i = i=1 n d G (y j = j=1 l k n d + G (y j = j=1 n d + G (x i = i=1 n p i = n(n 1 k i=1 n p i = n(n 1 l i=1 n (n 1 kq j, j=1 n (n 1 kp i. i=1 n q j, (1 j=1 n q j. ( j=1 l

51 Vol. 8 Distar decompositions of a symmetric product digraph 45 Since k = l (1 implies k( n i=1 p i + n j=1 q j = n(n 1 Thus k n(n 1. This completes (ii. Consider (iii. By assumption, k l. Then (1 and ( imply (l k n i=1 p i = (l k n j=1 q j n i=1 p i = n j=1 q j, Thus by (1, Thus (k + l n(n 1. n p i = i=1 n q j = n(n 1 k + l, (3 j=1 Note that the number of distars in D with centers in M is n(n 1 (k + l and so is the number of distars in D with centers in N. Thus there is a vertex x i in M which is the center of atleast n 1 k+l distars. Hence n 1 k + l l d C (x n,n 1 i = n 1, n 1 k + l k d+ C (x n,n 1 i = n 1. This implies n 1 k + l min{ n 1 1, n }. (4 k l Similarly, (4 follows if there is a vertex y j in N which is also the center of atleast n 1 k+l distars. This completes (iii. The proof of (iv is obvious. Since gcd(n, n 1 = 1 and k + l = n 1. Theorem.. If k n 1 and k (n 1, then C n,n 1 has an S k,k -decomposition. Proof. By the assumption, n 1 = tk for some integer t. For j = 1,,..., n, let H j be the sub digraph of C n,n 1 induced by M {y j }. Then H j is a directed multiple star with center y j and d + H j (y j = d H j (y j = n 1 = kt and e Hj (y j, x i = t e Hj (y j, x j = 0 t (for every x i M and i j. Thus, by Lemma 1.1, H j has an S k,k -decomposition. Since C n,n 1 can be decomposed into H 1, H,..., H n. C n,n 1 has an S k,k -decomposition. Corollary.1. Suppose k + l n 1. Then C n,n 1 has an S k,l -decomposition if and only if k = l and k (n 1. Theorem.3. If k + l = n 1 and k l, then C n,n 1 has S k,l -decomposition. Proof. We consider C n,n 1 as the symmetric digraph with vertex set {x 1, x,..., x n } {y 1, y,..., y n }. Let the distar S k,l decomposition of C n,n 1 be as follows: For 1 i n, T i = {(y j+1, x i : i j i + l 1} {(x i, y j+l : i + 1 j i + k}, T n+i = {(y i, x j+1 : i j i + k 1} {(x j+k, y i : i + 1 j i + l} where the subscripts of x and y are taken modulo n. As an example, the distar S 3, -decomposition of C 6,5 are given in Fig. 1. x 1 y 4 y y 5 y 6 T 1 y 3 x y 4 y 5 y6 y 1 y 3 T x 3 y 5 y 6 y y 1 y 4 T 3

52 46 V. Maheswari and A. Nagarajan No. 4 x 4 y 1 y y y 5 3 y T 4 1 y 6 x x x 3 x 5 4 y 4 T 7 x 6 x 5 x x 6 x 1 T 10 x 3 x 5 y 1 y y3 y 4 y 6 T 5 y x 1 x 3 x4 x 5 x 6 T 8 y 5 x 4 x 6 x1 x x 3 T 11 Fig. 1 x 6 y y 3 y y 4 y 1 5 y 3 T 6 x x 4 x x 5 x 1 6 y 6 T 9 x 5 x 1 x x x 4 3 T 1 Corollary.. Suppose k + l = n 1 and k l, then Cn,n 1 has an S k,l -decomposition if and only if (k + l n(n 1. Lemma.1. Cn,n 1 has an S k,l -decomposition if and only if there exists a non-negative integral function f defined on V (Cn,n 1, and Cn,n 1 contains a spanning sub digraph G such that (i for every u V (Cn,n 1, d G (u = lf(u and d+ G (u = d+ C kf(u. n,n 1(u (ii iff(u = 1 and (v, u E(G then (u, v E(G. Proof. (Necessity Suppose D is an S k,l -decomposition of Cn,n 1. For a vertex u, let f(u be the number of distars in D with center at u. Then f is a non-negative integral function defined on V (Cn,n 1. Let G be the spanning sub digraph of Cn,n 1 of which the arcs are those of distars in D which are oriented to their centers. Then d G (u = lf(u and d+ G (u = d+ C kf(u. n,n 1(u Suppose f(u = 1 and (v, u E(G. Then (v, u is an arc of a distar in D with center at v. Thus (u, v E(G. (Sufficiency For each u V (Cn,n 1, let H(u be the directed multiple star with center at u and arc set {(v, u : (v, u E(G} {(u, w : (u, w / E(G}. The center of H(u has in degree lf(u and out degree kf(u. If f(u, then e H(u (v, u e C n,n 1 (v, u = f(u. If f(u = 1, then (u, v and (v, u cannot both in H(u. Hence e H(u (v, u 1 = f(u. Thus, by Lemma 1.1, H(u has an S k,l -decomposition. Since {H(u : u V (Cn,n 1} partitions E(Cn,n 1, Cn,n 1 has an S k,l -decomposition. Theorem.4. Suppose k, l, n are positive integers such that k + l n 1, k l, (k + l n(n 1, n 1 n 1 n 1 n 1 k+l and k+l min{ k, l }. Then Cn,n 1 has an S k,l - decomposition. Proof. Let p = n 1 k+l and n = n(n 1 k+l np. Note that n 0. Define the numbers a i, b i, c i, d i (1 i n as follows: If n = 0, let a i = c i = pl if 1 i n, b i = d i = n 1 pk

53 Vol. 8 Distar decompositions of a symmetric product digraph 47 if 1 i n. If n 1, let (p + 1l, if 1 i n, a i = c i = pl, if n + 1 i n. n 1 (p + 1k, if 1 i n, b i = d i = n 1 pk, if n + 1 i n. Since n 1 k+l, p. Now apply the sufficiency of Lemma.1. By suitably choosing the non-negative integer function f, we can see that Cn,n 1 has an S k,l -decomposition if {((a 1, b 1, (a, b,..., (a n, b n, ((c 1, d 1, (c, d,..., (c n, d n } is bipartite digraphical. By Lemma 1., it suffices to show that {(a 1, a,..., a n, (d 1, d,..., d n } and {(b 1, b,..., b n, (c 1, c,..., c n } are both bigraphical. We first prove that {(a 1, a,..., a n, (d 1, d,..., d n } is bigraphical. and n a i = (p + 1ln + pl(n n i=1 = pln + ln + pln pln = pln + ln = l(n + np = ln(n 1 k + l n d i = (n 1 (p + 1kn + (n 1 pk(n n i=1 = (n 1n (p + 1kn + (n 1n + pkn npk (n 1n = kn + n(n 1 npk = n(n 1 (np + n k n(n 1 = n(n 1 k k + l n(n 1 = l. k + l Thus n i=1 a i = n i=1 d i. Next we show that r i=1 a i n i=1 min{d i, r} for 1 r n. We distinguish three cases depending on the relative position of r and the magnitudes of d i (i = 1,,..., n. We first evaluate r i=1 a i. For the case (k + l (n 1, we have p = n 1 k+l np = 0 and a i = pl for all 1 i n. Hence n = n(n 1 k+l, which implies r plr, if (k + l (n 1, a i = (p + 1lr, if (k + l (n 1 and r n, (p + 1lr (r n l, if (k + l (n 1 and n + 1 r n. i=1

54 48 V. Maheswari and A. Nagarajan No. 4 Case (1: r d i for all d i. Then for 1 r n, we have n min{d i, r} = i=1 = n i=1 n i=1 d i a i r a i. This completes Case(1. In the following, we define, for r = 1,,..., n, (G = n i=1 min{d i, r} r i=1 a i. It suffices to show each (r 0. Case (: r d i for all d i. Then n i=1 min{d i, r} = n i=1 r = nr. (n plr, if (k + l (n 1, (r = (n (p + 1lr, if (k + l (n 1 and r n, (n (p + 1lr + (r n l, if (k + l (n 1 and n r n. i=1 Note that n 1 k + l = p, if (k + l (n 1, p + 1, if (k + l (n 1. By the assumption that n 1 k+l n 1 l min{ n 1 k, n 1 }, we have l p, if (k + l (n 1, p + 1, if (k + l (n 1. Thus n 1 pl 0 if (k + l (n 1, and n 1 (p + 1l 0 if (k + l (n 1. This implies (r > 0, which completes Case (. Case (3: Neither r d i for all d i nor r d i for all d i. Then n 1 (p+1k < r < n 1 pk, and n i=1 min{d i, r} = (n 1 (p + 1kn + (n n r. Let (r be defined as in Case(. Thus (n 1 (p + 1kn + (n n plr, if (k + l (n 1, (n 1 (p + 1kn + (r = (n n (p + 1lr, if (k + l (n 1 and r n, (n 1 (p + 1kn + (n n (p + 1lr + (r n l, if (k + l (n 1 and n r n. Since r > n 1 (p + 1k, we have (n 1 (p + 1k(n pl, if (k + l (n 1, (r > (n 1 (p + 1k(n (p + 1l, if (k + l (n 1 and r n, (n 1 (p + 1k(n (p + 1l + (r n l, if (k + l (n 1 and n r n.

55 Vol. 8 Distar decompositions of a symmetric product digraph 49 Note that in this case n 0. Thus (k + l (n 1 and p + 1 = n 1 k+l. By the assumption that n 1 n 1 k+l min{ k, n 1 l }, we have n 1 k n 1 k p + 1, which implies (n 1 (p + 1k 0. Note that (3 also holds in Case (3. Thus (r > 0. This completes Case (3. By Theorem 1.1, {(a 1, a,..., a n, (d 1, d,..., d n } is bigraphical. Similarly, {(b 1, b,..., b n, (c 1, c,..., c n } is bigraphical. This completes the proof. Theorem.5. If k = l = n 1, then (P 3 K n has S k,l -decomposition. Proof. Let V (P 3 = {x 1, x, x 3 } and V (K n = {y 1, y,... y n }. Then V ((P 3 K n = 3 i=1 {x i V (K n } = V 1 V V 3 and V i = n j=1 {xj i }, 1 i 3, where xj i stands for (x i, y j. The distar S k,l -decomposition of (P 3 K n is given as follows: For 1 k n, T k = {(x j 1, xk, x j 3 : 1 j n and j k} and T k = {(xj 3, xk, x j 1 : 1 j n and j k}. Theorem.6. Suppose (P 3 K n has an S k,l -decomposition and if n k + l (n 1, then k = l and k (n 1. Proof. Let D be an S k,l - decomposition of (P 3 K n. Let V ((P 3 K n = X Z Y. For i = 1,,..., n, let r i be the number of distars S k,l in D with centers z i. Since n k + l (n 1, any distar S k,l in D must have its center in Z. Hence (n 1 = d (z i = lr i, (n 1 = d + (z i = kr i Thus k = l and k (n 1. Corollary.3. Suppose n k + l (n 1. Then (P 3 K n has S k,l -decomposition if and only if k = l = n 1 and k (n 1. Theorem.7. (P m K n has S k,l -decomposition if any one of the following conditions hold (i If k + l n 1, k = l and k n 1. (ii If k + l = n 1 and k l. (iii If k + l n 1, k l, (k + l n(n 1, n 1 k+l (iv If k = l = n 1 and m is odd. and n 1 k+l min{ n 1 k, n 1 }. Proof. For (i, (ii and (iii, we have C n,n 1 has S k,l -decomposition. Also we know that (P m K n = Cn,n 1 Cn,n 1... Cn,n 1. }{{} (m 1times Thus (P m K n has S k,l -decomposition. Similarly, if condition (iv holds then (P 3 K n has S k,l -decomposition. Also (P m K n = (P 3 K n... (P 3 K n. }{{} m 1 times Hence (P m K n has S k,l -decomposition for this case also. Thus the result follows. l References [1] P. V. Caetano and K. Heinrich, A note on distar factorizations, Ars Combinatoria, 30(1990, 7-3. [] G. Chartrand and L. Lesniak, Graphs and Digraphs, Fourth Edition, CRC Press, Boca Raton, 004.

56 50 V. Maheswari and A. Nagarajan No. 4 [3] C. J. Colbourn, D. G. Hoffman and C. A. Rodger, Directed star decompositions of the complete directed graph, J. Graph Theory, 16(199, [4] C. J. Colbourn, D. G. Hoffman and C. A. Rodger, Directed star decompositions of directed multigraphs, Discrete Math, 97(1991, [5] Hung-Chih Lee, Balanced Decompositions of Graphs and Factorizations of Digraphs, Ph. D Thesis, National Central University. [6] H. J. Ryser, Combinatorial properties of matrices of zeros and ones, Canad. J. Math, 9(1957,

57 Scientia Magna Vol. 8 (01, No. 4, The generalized difference gai sequences of fuzzy numbers defined by orlicz functions N. Subramanian, S. Krishnamoorthy and S. Balasubramanian Department of Mathematics, SASTRA University, Tanjore , India Department of Mathematics, Govenment Arts College (Autonomus, Kumbakonam, 61001, India nsmaths@yahoo.com drsk 01@yahoo.com sbalasubramanian009@yahoo.com Abstract In this paper we introduce the classes of gai sequences of fuzzy numbers using generalized difference operator m (m fixed positive integer and the Orlicz functions. We study its different properties and also we obtain some inclusion results these classes. Keywords Fuzzy numbers, difference sequence, Orlicz space, entire sequence, analytic sequ -ence, gai sequence, complete. 000 Mathematics subject classification: 40A05, 40C05, 40D Introduction The concept of fuzzy sets and fuzzy set operations were first introduced Zadeh [18] and subsequently several authors have discussed various aspects of the theory and applications of fuzzy sets such as fuzzy topological spaces, similarity relations and fuzzy orderings, fuzzy measures of fuzzy events, fuzzy mathematical programming. In this paper we introduce and examine the concepts of Orlicz space of entire sequence of fuzzy numbers generated by infinite matrices. Let C (R n = {A R n : A compact and convex}. The space C (R n has linear structure induced by the operations A + B = {a + b : a A, b B} and λa = {λa : a A} for A, B C (R n and λ R. The Hausdorff distance between A and B of C (R n is defined as δ (A, B = max {sup a A inf b B a b, sup b B inf a A a b }. It is well known that (C (R n, δ is a complete metric space. The fuzzy number is a function X from R n to [0,1] which is normal, fuzzy convex, upper semi-continuous and the closure of {x R n : X(x > 0} is compact. These properties imply that for each 0 < α 1, the α-level set [X] α = {x R n : X(x α} is a nonempty compact convex subset of R n, with support X c = {x R n : X(x > 0}. Let L (R n denote the set of all fuzzy numbers. The linear structure of L (R n induces the addition X + Y and scalar multiplication λx, λ R, in terms of α-level sets, by X + Y α = X α + Y α, λx α = λ X α for each 0 α 1. Define, for each 1 q <,

58 5 N. Subramanian, S. Krishnamoorthy and S. Balasubramanian No. 4 d q (X, Y = ( 1 0 δ (X α, Y α q dα 1/q and d = sup 0 α 1 δ (X α, Y α, where δ is the Hausdorff metric. Clearly d (X, Y = lim q d q (X, Y with d q d r, if q r [11]. Throughout the paper, d will denote d q with 1 q. The additive identity in L (R n is denoted by 0. For simplicity in notation, we shall write throughout d instead of d q with 1 q. A metric on L (R n is said to be translation invariant if d (X + Z, Y + Z = d (X, Y for all X, Y, Z L (R n A sequence X = (X k of fuzzy numbers is a function X from the set N of natural numbers into L (R n. The fuzzy number X k denotes the value of the function at k N. We denotes by W (F the set of all sequences X = (X k of fuzzy numbers. A complex sequence, whose k th terms is x k is denoted by {x k } or simply x. Let φ be the set of all finite sequences. Let l, c, c 0 be the sequence spaces of bounded, convergent and null sequences x = (x k respectively. In respect of l, c, c 0 we have x = sup k x k, where x = (x k c 0 c l. A sequence x = {x k } is said to be analytic if sup k x k 1/k <. The vector space of all analytic sequences will be denoted by Λ. A sequence x is called entire sequence if lim k x k 1/k = 0. The vector space of all entire sequences will be denoted by Γ. A sequence x is called gai sequence if lim k (k! x k 1/k = 0. The vector space of all gai sequences will be denoted by χ. Orlicz [6] used the idea of Orlicz function to construct the space (L M. Lindenstrauss and Tzafriri [7] investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space l M contains a subspace isomorphic to l p (1 p <. Subsequently different classes of sequence spaces defined by Parashar and Choudhary [8], Mursaleen [9], Bektas and Altin [30], Tripathy [31], Rao and subramanian [3] and many others. The Orlicz sequence spaces are the special cases of Orlicz spaces studied in Ref [33]. Recall [6,33] an Orlicz function is a function M : [0, [o, which is continuous, non-decreasing and convex with M(0 = 0, M(x > 0, for x > 0 and M(x as x. If convexity of Orlicz function M is replaced by M(x + y M(x + M(y then this function is called modulus function, introduced by Nakano [34] and further discussed by Ruckle [35] and Maddox [36] and many others. An Orlicz function M is said to satisfy -condition for all values of u, if there exists a constant K > 0, such that M(u KM(u (u 0. The -condition is equivalent to M(lu KlM(u, for all values of u and for l > 1. Lindenstrauss and Tzafriri [7] used the idea of Orlicz function to construct Orlicz sequence space { ( } xk l M = x w : M <, for some ρ > 0. (1 ρ The space l M with the norm x = inf k=1 { ρ > 0 : ( } xk M 1, ( ρ k=1 becomes a Banach space which is called an Orlicz sequence space. For M(t = t p, 1 p <, the space l M coincide with the classical sequence space l p Given a sequence x = {x k } its n th

59 Vol. 8 The generalized difference gai sequences of fuzzy numbers defined by orlicz functions 53 section is the sequence x (n = {x 1, x,..., x n, 0, 0,...}, δ (n = (0, 0,..., 1, 0, 0,..., 1 in the n th place and zero s else where.. Remark An Orlicz function M satisfies the inequality M (λx λm (x for all λ with 0 < λ < 1. Let m N be fixed, then the generalized difference operation is defined by m : W (F W (F X k = X k X k+1 and m X k = ( m 1 X k (m for all k N. 3. Definitions and preliminaries Let P s denotes the class of subsets of N, the natural numbers, those do not contain more than s elements. Throughout (φ n represents a non-decreasing sequence of real numbers such that nφ n+1 (n + 1 φ n for all n N. The sequence χ (φ for real numbers is defined as follows { } 1 χ (φ = (X k : φ s (k! X k 1/k 0 as k, s for k σ P s. The generalized sequence space χ ( n, φ of the sequence space χ (φ for real numbers is defined as follows { } 1 χ ( n, φ = (X k : φ s (k! X k 1/k 0 as k, s for k σ P s, where n X k = X k X k+n for k N and fixed n N. In this article we introduce the following classes of sequences of fuzzy numbers. Let M be an Orlicz function, then = = = Λ F M ( m (X k W (F : sup k (( M d m X k 1/k, 0 <, for some ρ > 0 ρ. χ F M ( m ( (X k W (F : M d (k! m X k 1/k, 0 0 as k, for some ρ > 0 ρ. Γ F M ( m (( (X k W (F : M d m X k 1/k, 0 0 as k, for some ρ > 0 ρ.

60 54 N. Subramanian, S. Krishnamoorthy and S. Balasubramanian No. 4 = = χ F M ( m, φ ( (X 1 k W (F : M d (k! m X k 1/k, 0 0 as k, s, for k σ P s φ s ρ. Γ F M ( m, φ ( (X 1 k W (F : M d ( m X k 1/k, 0 0 as k, s, for k σ P s φ s ρ. 4. Main results In this section we prove some results involving the classes of sequences of fuzzy numbers χ F M ( m, φ, χ F M ( m and Λ F M ( m. Theorem 4.1. If d is a translation invariant metric, then χ F M ( m, φ are closed under the operations of addition and scalar multiplication. Proof. Since d is a translation invariant metric implies that ( ( ( d (k! ( m X k + m Y k 1/k, 0 d (k! ( m X k 1/k, 0 + d (k! ( m Y k 1/k, 0 (3 and ( ( d (k! ( m λx k 1/k, 0 λ 1/k d (k! ( m X k 1/k, 0, (4 where λ is a scalar and λ 1/k > 1. Let X = (X k and Y = (Y k χ F M ( m, φ. Then there exist positive numbers ρ 1 and ρ such that ( 1 d((k! φ s M m X k 1/k, 0 ρ 1 0 as k, s, for k σ P s. ( 1 d((k! φ s M m X k 1/k, 0 ρ 0 as k, s, for k σ P s. Let ρ 3 = max (ρ 1, ρ. By the equation (3 and since M is non-decreasing convex function, we have ( M d (k! m X k + m Y k 1/k, 0 ρ ( M d ( (k! m X k 1/k, 0 + d (k! m Y k 1/k, 0 1 M ρ 3 ( d (k! m X k 1/k, 0 ρ 1 ( 1 M d (k! m X k + m Y k 1/k, 0 φ s ρ 1 ρ 3 ρ 3 ( + 1 M d (k! m Y k 1/k, 0 ( 1 M d ( (k! m X k 1/k, M d (k! m Y k 1/k, 0. φ s φ s ρ ρ

61 Vol. 8 The generalized difference gai sequences of fuzzy numbers defined by orlicz functions 55 for k σ P s, Hence X + Y χ F M ( m, φ. Now, let X = (X k χ F M ( m, φ and λ R with 0 < λ 1/k < 1. By the condition (4 and Remark, we have ( M d (k! m λx k 1/k, 0 ρ ( M λ 1/k d (k! m X k 1/k, 0 ρ ( λ 1/k M d (k! m X k 1/k, 0. ρ Therefore λx χ F M ( m, φ. This completes the proof. Theorem 4.. The space χ F M ( m, φ is a complete metric space with the metric by g (X, Y = d ((k! X k Y k 1/k ( + inf ρ > 0 : sup 1 M d k! ( m X k m Y k 1/k 1 k σ P s φ s ρ. Proof. Let ( X i be a cauchy sequence in χ F M ( m, φ. Then for each ɛ > 0, there exists a positive integer n 0 such that g ( X i, Y j < ɛ for i, j n 0, then ( ( { ( d k! Xk i Y j 1/k d (k!( 1 k +inf ρ > 0 : sup k σ Ps φ s (M m X i k m Y j k 1/k } ρ 1 < ɛ, for all i, j n 0 ( ( d k! Xk i Y j 1/k k < ɛ ( { 1 for all i, j n 0 and inf ρ > 0 : sup k σ Ps φ s (M } d(k!( m X k m Y k 1/k ρ 1 < ɛ for all i, j n 0. ( ( By (5, d k! Xk i 1/k Xj k < ɛ for all i, j n 0 and k = 1,, 3,..., m. It follows that ( ( X i k is a cauchy sequence in L (R for k = 1,, 3,..., m. Since L (R is complete, then X i k is convergent in L (R. Let lim i Xk i = X k for k = 1,,..., m. Now (6 for a given ɛ > 0, there exists some ρ ɛ (0 < ρ ɛ < ɛ such that ( d (k!( 1 sup k σ Ps φ s (M m X i k m Xk j 1/k ρ ɛ 1. Thus sup 1 k σ P s φ s sup 1 k σ P s φ s M M ( d k! ( d k! ( m Xk i m X j 1/k k ρ 1 ( m Xk i m X j 1/k k 1, ρ ɛ

62 56 N. Subramanian, S. Krishnamoorthy and S. Balasubramanian No. 4 ( ( we have d k! m Xk i m X j 1/k k < ɛ and the fact that ( ( d k! Xk+m i X j 1/k k+m ( ( d k! m Xk i m X j 1/k k = m ( ( d k! Xk i X j 1/k k 0 + m ( ( d k! Xk+1 i X j 1/k k m ( ( d k! Xk+m 1 i X j 1/k k+m 1. m 1 ( ( So, we have d k! Xk i 1/k Xj k < ɛ for each k N. Therefore ( X i is a cauchy sequence in L (R. Since L (R. is complete, then it is convergent in L (R. Let lim i X i k = X k say, for each k N. Since ( X i is a cauchy sequence, for each ɛ > 0, there exists n 0 = n 0 (ɛ such that g ( X i, X j < ɛ for all i, j n 0. So we have ( ( lim d k! Xk i X j 1/k j k ( (k! = d Xk i X k 1/k < ɛ and ( ( lim d k! m Xk i m X j 1/k j k ( (k! = d m Xk i m X k 1/k < ɛ for all i, j n 0. This implies that g ( X i, X < ɛ for all i n 0. That is X i X as i, where X = (X k. Since ( d (k! m X k X 0 1/k ( d (k! m X n0 k X 0 1/k ( + d (k! m X n0 k m X k 1/k, we obtain X = (X k χ F M. Therefore χf M ( m, φ is complete metric space. This completes the proof. Proposition 4.1. The space Λ F M ( m is a complete metric space with the metric by h (X, Y = inf {ρ > 0 : sup k ( M ( d(( m X k m Y k 1/k ρ } 1. ( Theorem 4.3. If φs ψ s as s then χ F M ( m, φ χ F M ( m, ψ ( Proof. Let φs ψ s 0 as s and X = (X k χ F M ( m, φ. Then, for some ρ > 0, ( 1 M d (k! m X k 1/k, 0 0 as k, s, for k σ P s φ s ρ ( 1 M d (k! m X k 1/k, 0 ( ( φs 1 M d (k! m X k 1/k, 0. ψ s ρ ψ s φ s ρ

63 Vol. 8 The generalized difference gai sequences of fuzzy numbers defined by orlicz functions 57 Therefore X = (X k χ F M ( m, ψ. Hence χ F M ( m, φ χ F M ( m, ψ. This completes the proof. Proposition 4.. If ( ( φs ψs and 0 as s, ψ s φ s then χ F M ( m, φ = χ F M ( m, ψ. Theorem 4.4. χ F M ( m Γ F M ( m, φ. Proof. Let X = (X k χ F M ( m. Then we have ( M 0 as k for some ρ > 0. d((k! m X k 1/k, 0 ρ Since (φ n is monotonic increasing, so we have ( 1 M d (k! m X k 1/k, 0 φ s ρ ( 1 M d (k! m X k 1/k, 0 φ 1 ρ ( 1 M d (k! m X k 1/k, 0. φ s ρ Therefore for k σ P s. Hence ( 1 M d (k! m X k 1/k, 0 0 as k, s φ s ρ ( 1 M d ( m X k 1/k, 0 0 as k, s φ s ρ for k σ P s and (k! 1/k 1. Thus X = (X k Γ F M ( m, φ. Therefore χ F M ( m Γ F M ( m, φ. This completes the proof. Theorem 4.5. Let M 1 and M be Orlicz functions satisfying -condition. Then χ F M ( m, φ χ F M 1 M ( m, φ. Proof. Let X = (X k Γ F M ( m, φ. Then there exists ρ > 0 such that

64 58 N. Subramanian, S. Krishnamoorthy and S. Balasubramanian No. 4 ( 1 d((k! φ s M m X k 1/k, 0 ρ 0 as k, s for k σ P s. Let 0 < ɛ < 1 and δ with 0 < δ < 1 such that M 1 (t < ɛ for 0 1 < δ. Let ( y k = M for all k N. d((k! m X k 1/k, 0 ρ Now, let M 1 (y k = M 1 (y k + M 1 (y k, (5 where the equation (7 RHS of the first term is over y k δ and the equation of (7 RHS of the second term is over y k > δ. By the Remark, we have For y k > δ, y k < y k δ 1 + y k δ. Since M 1 is non-decreasing and convex, so Since M 1 satisfies -condition, so M 1 (y k M 1 (1 y k + M 1 ( y k. (6 ( M 1 (y k < M y k < 1 δ M 1 ( + 1 M 1 M 1 (y k < 1 KM 1 ( y k δ + 1 KM 1 ( y k δ. Hence the equation (7 in RHS of second terms is By equation (8 and (9, we have ( yk. δ M 1 (y k max ( 1, Kδ 1 M 1 ( y k. (7 X = (X k χ F M 1 M ( m, φ. Thus, χ F M ( m, φ χ F M 1 M ( m, φ. This completes the proof. Propositon 4.3. Let M be an Orlicz function which satisfies -condition. Then χ F ( m, φ χ F M ( m, φ References [1] Aytar, S. Aytar, Statistical limit points of sequences of fuzzy numbers, Inform. Sci., 165(004, [] Basarir, M. Basarir and M. Mursaleen, Some sequence spaces of fuzzy numbers generated by infinite matrices, J. Fuzzy Math., 11(003, No. 3, [3] Bilgin, T. Bilgin, -statistical and strong -Cesáro convergence of sequences of fuzzy numbers, Math. Commun., 8(003,

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67 Scientia Magna Vol. 8 (01, No. 4, Hankel determinant for a new subclass of analytic functions Gagandeep Singh Department of Mathematics, DIPS College (Co-Ed., Dhilwan (Kapurthala, Punjab, India kamboj.gagandeep@yahoo.in Abstract We introduce a new subclass of analytic-univalent functions and determine the sharp upper bounds of the second Hankel determinant for the functions belonging this class. Keywords Analytic functions, starlike functions, convex functions, Hankel determinant, coefficient bounds. 1. Introduction and preliminaries Let A be the class of analytic functions of the form f(z = z + a k z k (1 in the unit disc E = {z : z < 1}. Let S be the class of functions f(z A and univalent in E. Let M(α (0 α 1 be the class of functions f(z A which satisfy the condition Re { k= zf (z + αz f (z (1 αf(z + αzf (z } > 0, z E. ( This class was introduced by Singh [13] and Fekete-Szegö inequality for functions of this class was establihed by him. Obviously M(α is the subclass of the class of α-convex functions introduced by Mocanu [9]. In particular M(0 S, the class of starlike functions and M(1 K, the class of convex functions. In 1976, Noonan and Thomas [10] stated the q th Hankel determinant for q 1 and n 1 as a n a n+1... a n+q 1 a n H q (n = a n+q a n+q This determinant has also been considered by several authors. For example, Noor [11] determined the rate of growth of H q (n as n for functions given by Eq. (1 with bounded

68 6 Gagandeep Singh No.4 boundary. Ehrenborg [1] studied the Hankel determinant of exponential polynomials and the Hankel transform of an integer sequence is defined and some of its properties discussed by Layman [5]. Also Hankel determinant for various classes was studied by several authors including Hayman [3], Pommerenke [1] and recently by Mehrok and Singh [8]. Easily, one can observe that the Fekete-Szegö functional is H (1. Fekete and Szegö [] then further generalised the estimate of a 3 µa where µ is real and f S. For our discussion in this paper, we consider the Hankel determinant in the case of q = and n =, a a 3 a 3 a 4. In this paper, we seek upper bound of the functional a a 4 a 3 for functions belonging to the above defined class.. Main result Let P be the family of all functions p analytic in E for which Re(p(z > 0 and for z E. Lemma.1. If p P, then p k (k = 1,, 3,... This result is due to Pommerenke [1]. Lemma.. If p P, then p(z = 1 + p 1 (z + p z +... (3 p = p 1 + (4 p 1x, 4p 3 = p p 1 (4 p 1x p 1 (4 p 1x + (4 p 1(1 x z for some x and z satisfying x 1, z 1 and p 1 [0, ]. This result was proved by Libera and Zlotkiewiez [6,7]. Theorem.1. If f M(α, then Proof. As f M(α, so from ( a a 4 a 3 1 (1 + α(1 + 3α. (4 zf (z + αz f (z = p(z. (5 (1 αf(z + αzf (z On expanding and equating the coefficients of z, z and z 3 in (5, we obtain a = p α, (6 a 3 = p (1 + α + p 1 (1 + α (7

69 Vol.8 Hankel determinant for a new subclass of analytic functions 63 and a 4 = p 3 3(1 + 3α + p 1p (1 + 3α + p 3 1 6(1 + 3α. (8 Using (6, (7 and (8, it yields [ ] a a 4 a 3 = 1 4(1 + α p 1 p 3 + 6((1 + α (1 + α(1 + 3αp 1p, (9 C(α +((1 + α 3(1 + α(1 + 3αp 4 1 3(1 + α(1 + 3αp where C(α = 1(1 + α(1 + 3α(1 + α. Using Lemma.1 and Lemma. in (9, we obtain [ 4(1 + α 1α + 4(1 + 4α + α + 3(1 + α(1 + 3α]p 4 1 a a 4 a 3 = 1 +[8(1 + α + 1α 6(1 + α(1 + 3α]p 1(4 p 1x. 4C(α [(4(1 + α 3(1 + α(1 + 3αp 1 + 1(1 + α(1 + 3α](4 p 1x +8(1 + α p 1 (4 p 1(1 x z Assume that p 1 = p and p [0, ], using triangular inequality and z 1, we have [ 4(1 + α 1α + 4(1 + 4α + α + 3(1 + α(1 + 3α]p 4 a a 4 a 3 1 +[8(1 + α + 1α 6(1 + α(1 + 3α]p (4 p x 4C(α +[(4(1 + α 3(1 + α(1 + 3αp + 1(1 + α(1 + 3α](4 p x +8(1 + α p(4 p (1 x or a a 4 a 3 1 4C(α [ 4(1 + α 1α + 4(1 + 4α + α + 3(1 + α(1 + 3α]p 4 + 8(1 + α p(4 p + [8(1 + α + 1α 6(1 + α(1 + 3α]. p (4 p δ + [(4(1 + α 3(1 + α(1 + 3αp 8(1 + α p + 1(1 + α(1 + 3α](4 p δ Therefore where δ = x 1 and a a 4 a 3 = 1 4C(α F (δ, F (δ = [ 4(1 + α 1α + 4(1 + 4α + α + 3(1 + α(1 + 3α]p 4 +8(1 + α p(4 p + [8(1 + α + 1α 6(1 + α(1 + 3α]p (4 p δ +[(4(1 + α 3(1 + α(1 + 3αp 8(1 + α p + 1(1 + α(1 + 3α](4 p δ is an increasing function. Therefore MaxF (δ = F (1. Consequently where G(p = F (1. So a a 4 a 3 1 G(p, (10 4C(α G(p = A(αp 4 + B(αp + 48(1 + α(1 + 3α,

70 64 Gagandeep Singh No.4 where and Now and G (p = 0 gives A(α = 48α B(α = 96α. G (p = 4A(αp 3 + B(αp G (p = 1A(αp + B(α. p[a(αp B(α] = 0. G (p is negative at B(α p = A(α = 1. So MaxG(p = G(1. Hence from (10, we obtain (4. The result is sharp for p 1 = 1, p = p 1 and p 3 = p 1 (p 1 3. For α = 0 and α = 1 respectively, we obtain the following results due to Janteng [4]. Corollary.1. If f(z S, then Corollary.. If f(z K, then a a 4 a 3 1. a a 4 a References [1] R. Ehrenborg, The Hankel determinant of exponential polynomials, American Mathematical Monthly, 107(000, [] M. Fekete and G. Szegö, Eine Bemerkung über ungerade schlichte Funktionen, J. London Math. Soc., 8(1933, [3] W. K. Hayman, Multivalent functions, Cambridge Tracts in Math. and Math. Phys., No. 48, Cambridge University Press, Cambridge, [4] Aini Janteng, Suzeini Abdul Halim and Maslina Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal., 1(007, No. 13, [5] J. W. Layman, The Hankel transform and some of its properties, J. of Integer Sequences, 4(001, [6] R. J. Libera and E-J. Zlotkiewiez, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc., 85(198, [7] R. J. Libera and E-J. Zlotkiewiez, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 87(1983,

71 Vol.8 Hankel determinant for a new subclass of analytic functions 65 [8] B. S. Mehrok and Gagandeep Singh, Estimate of second Hankel determinant for certain classes of analytic functions, Scientia Magna, 8(01, No. 3, [9] P. T. Mocanu, Une propriete de convexite généralisée dans la théorie de la représentation conforme, Mathematica(CLUJ, 11(1969, No. 34, [10] J. W. Noonan and D.K.Thomas, On the second Hankel determinant of a really mean p-valent functions, Trans. Amer. Math. Soc., 3(1976, No., [11] K. I. Noor, Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roum. Math. Pures Et Appl., 8(1983, No. 8, [1] Ch. Pommerenke, Univalent functions, Göttingen: Vandenhoeck and Ruprecht., [13] Gagandeep Singh, Fekete-Szegö coefficient functional for certain classes of analytic functions, To appear.

72 Scientia Magna Vol. 8 (01, No. 4, On non-unit speed curves in Minkowski 3-space Yasin Ünlütürk and Ümit Ziya Savcı Department of Mathematics Kırklareli University, 39100, Kırklareli-Turkey Department of Mathematics Karabük University, 78050, Karabük-Turkey yasinunluturk@kirklareli.edu.tr zsavci@hotmail.com Abstract In this work, by means of the method used to obtain Frenet apparatus and Darboux frame for non-unit speed curve in E 3 in [3], we present Frenet apparatus for non-unit speed curves in E 3 1. Also, Darboux frame for non-unit speed curves and surface pair is given in E 3 1. Keywords Non-unit speed curve, spacelike non-unit speed curve, timelike non-unit speed c -urve, Darboux frame on non-unit speed curve-surface pair. 000 Mathematics Subject Classification: 53A05, 53B5, 53B Introduction Suffice it to say that the many important results in the theory of the curves in E 3 were initiated by G. Monge and the moving frames idea was due to G. Darboux. In the classical differential geometry of curves in E 3, the unit speed curve which is obtained by using the norm of the curve, is a well known concept. In the light of this idea, there are many studies on the unit speed curve in Minkowski space. Frenet apparatus for non-unit speed curve E 3 has been studied by Sabuncuoglu in [3]. Also he has studied Darboux frame and its derivative formulas for non-unit speed curves and surface pair in E 3 [3]. In this work, by means of the method used to obtain Frenet apparatus and Darboux frame for non-unit speed curves in E 3 in [3], we present Frenet apparatus for non-unit speed curves and Darboux frame and its derivative formulas for non-unit speed curves and surface pairs in Minkowski 3-space E Preliminaries Let E 3 1 be the three-dimensional Minkowski space, that is, the three-dimensional real vector space E 3 with the metric < dx,dx >=dx 1 + dx dx 3, where (x 1, x, x 3 denotes the canonical coordinates in E 3. An arbitrary vector x of E 3 1 is said to be spacelike if < x, x >>0 or x = 0, timelike if < x, x ><0 and lightlike or null if < x, x >=0

73 Vol. 8 On non-unit speed curves in Minkowski 3-space 67 and x= 0. A timelike or light-like vector in E 3 1 is said to be causal. For x E 3 1, the norm is defined by x = < x, x >, then the vector x is called a spacelike unit vector if < x, x >=1 and a timelike unit vector if < x, x >= 1. Similarly, a regular curve in E 3 1 can locally be spacelike, timelike or null (lightlike, if all of its velocity vectors are spacelike, timelike or null (lightlike, respectively []. For any two vectors x = (x 1, x, x 3 and y = (y 1, y, y 3 of E 3 1, the inner product is the real number < x, y >= x 1 y 1 + x y x 3 y 3 and the vector product is defined by x y = ((x y 3 x 3 y, (x 3 y 1 x 1 y 3, (x 1 y x y 1. The unit Lorentzian and hyperbolic spheres in E 3 1 are defined by H + = {x E 3 1 x 1 + x x 3 = 1, x 3 1}, S 1 = {x E 3 1 x 1 + x x 3 = 1}. (1 Let {T, n, b} be the moving Frenet frame along the curve α with arc-length parameter s. For a spacelike curve α, the Frenet Serret equations are T N B = 0 κ 0 T εκ 0 τ N 0 τ 0 B, ( where T, T = 1, N, N = ε = ±1, B, B = ε, T, N = T, B = N, B = 0. Furthermore, for a timelike non-unit speed curve α in E 3 1, the following Frenet formulae are given in as follows, T N B = 0 κ 0 T κ 0 τ N 0 τ 0 B where T, T = 1, N, N = B, B = 1, T, N = T, B = N, B = 0 [1]., (3 Let M be an oriented surface in three-dimensional Minkowski space E 3 1 and let consider a non-null curve α(t lying on M fully. Since the curve α(t is also in space, there exists Frenet frame {T, N, B} at each points of the curve where T is unit tangent vector, N is principal normal vector and B is binormal vector, respectively. Since the curve α(t lies on the surface M there exists another frame of the curve α(t which is called Darboux frame and denoted by {T, b, n}. In this frame T is the unit tangent of the curve, n is the unit normal of the surface M and b is a unit vector given by b = n T. Since the unit tangent T is common in both Frenet frame and Darboux frame, the vectors N, B, b and n lie on the same plane. Then, if the surface M is an oriented timelike surface, the relations between these frames can be given as follows. (i If the surface M is a timelike surface, then the curve α(t lying on M can be a spacelike or a timelike curve. Thus, the derivative formulae of the Darboux frame of α(t is given by T b n = 0 κ g εκ n T κ g 0 ετ g b κ n τ g 0 n, (4

74 68 Yasin Unluturk and Ümit Ziya Savcı No. 4 where T, T = ε = ±1, b, b = ε, n, n = 1. (ii If the surface M is a spacelike surface, then the curve α(t lying on M is a spacelike curve. Thus, the derivative formulae of the Darboux frame of α(t is given by T b n = where T, T = 1, b, b = 1, n, n = 1 [1]. 0 κ g κ n T κ g 0 τ g b κ n τ g 0 n, (5 In these formulae at (4 and (5, the functions κ n, κ g and τ g are called the geodesic curvature, the normal curvature and the geodesic torsion, respectively. Here and in the following, we use dot to denote the derivative with respect to the arc length parameter of a curve. Here, we have the following properties of α characterized by the conditions of κ g, κ n and τ g : (see []. α is a geodesic curve if and only if κ g = 0. α is a asymptotic curve if and only if κ n = 0. α is line of curvature if and only if τ g = Some properties of non-unit speed curves in E 3 1 Let α(t be an arbitrary curve with speed v = α = ds dt. Theoretically we may reparametrize α to get a unit speed curve α(s(t, so, we define curvature and torsion of α in terms of its arclength reparametrization α(s(t. Moreover, the tangent vector α (t is in the direction of the unit tangent T (s of the reparametrization, so T (s(t = α (t/ α (t. This says that we should define a non-unit speed curve s invariants in terms of its unit speed reparametrization s invariants. Definition 3.1. Let α be any non unit speed curve in Minkowski 3-space E 3 1, so (1 The unit tangent of α(t is defined to be T (t def = T (s(t. ( The curvature of α(t is defined to be κ(t def = κ(s(t. (3 If κ > 0, then the principal normal of α(t is defined to be N(t def = N(s(t. (4 If κ > 0, then the binormal of α(t is defined to be B(t def = B(s(t. (5 If κ > 0, then the torsion of α(t is defined to be τ(t def = τ(s(t. Theorem 3.1. If the curve α : I E 3 1 is a regular non-unit speed curve in E 3 1, then there is a parameter map h : I J so that (α h = 1 for all s J. Proof. Let the arc-length function of the curve α be f. It is known that f = α. The function f has values differed from zero since α is a regular curve. Hence f is a one to one function because it is an increasing function. Let s say f(i = J. Consequently, f becomes

75 Vol. 8 On non-unit speed curves in Minkowski 3-space 69 regular and bijective function. So has get the inverse, let h be the inverse of f and α h = β. So if we show that β : J E 3 1 is a unit speed curve, the proof is completed as follows: and h (s = (f 1 (s = β = (α h = h (α h = h α h, 1 f (h(s = 1 α (h(s = 1 (α h(s = 1 (α h (s. Theorem 3.. (The Frenet formulas for non-unit speed curves in E 3 1 For a regular curve α with speed v = ds dt, and curvature κ > 0, (i if α is a spacelike non-unit speed curve, then the derivative formula of Frenet frame is as follows: T = vκn, N = v( εκt + τb, B = vτn. (ii if α is a timelike non-unit speed curve, then the derivative formula of Frenet frame is as follows: T = vκn, N = v(κt + τb, B = vτn. Proof. (i For non-unit speed spacelike curve, the unit tangent T (t is T (s by the definition 3.1. Now T (t denotes differentiation with respect of t, so we must use the chain rule on the righthand side to determine κ and τ. From the definition 3.1, If we differentiate (6, the expression is obtained as T (t = T (s(t. (6 From (7, we get dt (t dt = dt (s(t ds = κ(sn(sv = κ(tn(tv. (7 ds dt T = κ(tvn(t. So the first formula of (i is proved. For the second and third, by the unit speed Frenet formulas, we have N (t = dn(s ds = ( κ(st (s + τ(sb(sv, ds dt N (t = κ(tvt (t + τ(tvb(t. And we get B (t = db(s ds = τ(sn(sv = τ(tvn(t. ds dt

76 70 Yasin Unluturk and Ümit Ziya Savcı No. 4 (ii For non-unit speed timelike curve, if the computations are made as follows: by the definition 3.1, T (t = T (s(t. (8 If we differentiate (8, the expression is found as From (9, we obtain dt (t dt = dt (s(t ds ds dt dt (s(t = v = vκ(tn(t. (9 ds T = κ(tvn(t. So the first formula of (ii for non-unit speed timelike curve is proved. For the second and third, we get N (t = dn(s ds = (κ(st (s + τ(sb(sv = (κ(tt (t + τ(tb(tv ds dt and finally B (t = db(s ds = τ(sn(sv = τ(tvn(t. ds dt Lemma 3.1. For a non-unit speed curve α in E 3 1, α = vt and α = dv dt T + κv N. Proof. Since α(t = α(s(t, the first calculation is while the second is α (t = α (s ds dt = vα (s = vt (s = vt, α (t = dv dt T (t + v(tt (t = dv dt T (t + κ(tv (tn(t. 4. Non-unit speed curve and surface pair in E 3 1 In this part, we will define the curvatures of curve-surface pair (α, M while α is a non-unit speed curve in E 3 1. Let s consider the non-unit speed curve α : I M. Previously, we proved that there is a parameter map h : J I so that α h : J M is a unit speed curve. The function h is the inverse of f(t = t t 0 α (r dr. Let s say α h = β. β is a unit speed curve. Let s the unit tangent vector field of β be denoted by T. The set {T, b, n β} is the frame of curve-surface pair (β, M, where b = (n β T. Let s J, h(s = t. s = f(t because of h(s = f 1 (s. So β(s = α(h(s = α(t. Definition 4.1. Let β be the unit speed curve obtained from the non-unit speed curve α : I M in E 3 1. While the frame of curve-surface pair (β, M is {T, b, n β}, the set {T, b, n α} defined by T (t = T (f(t, b(t = b(f(t, n(α(t = n(β(f(t is called Darboux frame of curve-surface pair (α, M in E 3 1. Also, while the curvatures of curvesurface pair (β, M are κ n, κ g, τ g, the curvatures of curve-surface pair (α, M are as follows: κ n (t = κ n (f(t, κ g (t = κ g (f(t, τ g (t = τ g (f(t.

77 Vol. 8 On non-unit speed curves in Minkowski 3-space 71 Theorem 4.1. Given the non-unit speed curve α : I M, the curvatures κ g, κ n and τ g of curve-surface pair (α, M in E 3 1 are as follows: κ n = 1 v α, n α, κ g = 1 v α, b, τ g = 1 v (n α, b. Proof. Let f(t = s, by definition 4.1, we get Let s calculate the components of (10, we write (11 as follows: If we differentiate (1, we get κ n (t = κ n (s = β (s, (n β(s. (10 β (s = T (s = T (t = 1 v(t α (t, (11 α (t = v(tβ (s = v(tβ (f(t = v(t(β f(t. (1 α (t = v (t(β f(t + v(tf (tβ (f(t = v (tβ (s + (v(t β (s. (13 By differentiating (11 and using (13, we get By using (14 in (10, β (s = 1 v (t (α (t v (tβ (s. (14 κ n (t = β (s, (n β(s 1 = v (t (α (t v (tβ (s, (n β(s = = = Let s see the equation of κ g (t. So 1 v (t { α (t, n β(s v (tβ (s, n β(s } 1 v (t α (t, n β f(t 1 v (t α (t, n α(t = ( 1 v α, n α (t. κ g (t = κ g (s = β (s, b(s. (15 By using (14 in (15, we get κ g (t = = = 1 v (t (α (t v (tβ (s, b(s 1 α v (t, b(f(t (t 1 v (t α (t, b(t.

78 7 Yasin Unluturk and Ümit Ziya Savcı No. 4 Finally, let s see the equation of τ g (t. So τ g (t = τ g (s = (n β (s, b(s = (n β (s, b(t. (16 Also we get (n α (t = (n β f (t = f (t(n β (f(t = v(t(n β (s or (n β (s = 1 v(t (n α (t. (17 By using (17 in (16, we get 1 τ g (t = v(t (n α (t, b(t = 1 v (n α, b (t. Theorem 4.. Given non-unit speed curve α : I M, (i If (α, M is timelike or spacelike curves on timelike surface pair, the derivative formula of Darboux frame {T, b, n α} is as follows: T (t = v(t[κ g (tb(t εκ n (t(n α(t], b (t = v(t[κ g (tt (t + ετ g (t(n α(t], (n α (t = v(t[κ n (tt (t + τ g (tb(t], for curve-surface pair (α, M in E 3 1. (ii If (α, M is spacelike curve on spacelike surface pair, the derivative formula of Darboux frame {T, b, n α} is as follows: T (t = v(t[κ g (tb(t + κ n (t(n α(t], b (t = v(t[ κ g (tt (t + τ g (t(n α(t], (n α (t = v(t[κ n (tt (t + τ g (tb(t], for curve-surface pair (α, M in E 3 1. Proof. (i For timelike surface: T (t = T (f(t = (T f(t. (18 By differentiating (18, we get T (t = (T f (t = f (tt (f(t = v(t[κ g (sb(s εκ n (s(n β(s] = v(t[κ g (tb(t εκ n (t(n α(t] and b(t = b(f(t = (b f(t. (19

79 Vol. 8 On non-unit speed curves in Minkowski 3-space 73 By differentiating (19, we get b (t = (b f (t = f (tb (f(t = v(t[κ g (st (s + ετ g (s(n β(s] = v(t[κ g (tt (t + ετ g (t(n α(t]. For the third formula of (i, (n α(t = (n β f(t. (10 By differentiating (10, we get (n α (t = (n β f (t = f (t(n β (f(t = v(t[κ n (st (s + τ g (sb(s] = v(t[κ n (tt (t + τ g (tb(t]. (ii For spacelike surface: T (t = T (f(t = (T f(t. (11 By differentiating (11, we get T (t = (T f (t = f (tt (f(t = v(t[κ g (sb(s + κ n (s(n β(s] = v(t[κ g (tb(t + κ n (t(n α(t] and b(t = b(f(t = (b f(t. (1 By differentiating (1, we get b (t = (b f (t = f (tb (f(t = v(t[ κ g (st (s + τ g (s(n β(s] = v(t[ κ g (tt (t + τ g (t(n α(t]. For the third formula of (ii, (n α(t = (n β f(t. (13 By differentiating (13, we get (n α (t = (n β f (t = f (t(n β (f(t = v(t[κ n (st (s + τ g (sb(s] = v(t[κ n (tt (t + τ g (tb(t].

80 74 Yasin Unluturk and Ümit Ziya Savcı No. 4 Corollary 4.1. Given non-unit speed curve α : I M, the curvatures of curve-surface pair (α, M in E 3 1 as follows: (i If (α, M is timelike or spacelike curves on timelike surface pair, then κ n = ε v T, n α, κ g = ε v T, b, τ g = ε v b, n α. (ii If (α, M is spacelike curve on spacelike surface pair, then κ n = 1 v T, n α, κ g = 1 v T, b, τ g = 1 v b, n α. Proof. The direct result of Theorem 4.3. References [1] R. Lopez, Differential geometry of curves and surfaces in Lorentz-Minkowski space, Mini-Course taught at the Instituto de Matematica e Estatistica (IME-USP, University of Sao Paulo, Brasil, 008. [] B. O Neill, Semi Riemannian geometry with applications to relativity, Academic Press, Inc. New York, [3] A. Sabuncuoğlu, Diferensiyel Geometri, Nobel Yayın Dağıtım, Ankara, 001, (in Turkish.

81 Scientia Magna Vol. 8 (01, No. 4, Self-integrating Polynomials in two variables A. C. F. Bueno Department of Mathematics and Physics, Central Luzon State University, Science City of Muñoz 310, Nueva Ecija, Philippines Abstract In this paper, a condition for self-integrability of a polynomial in two variables on a rectangular region R was obtained. Keywords Double integral, polynomial in two variables, rectangular region, self-integrability, self-integrable. 1. Introduction and preliminaries Graham [1] shown that the polynomials of the form p k (x = x k + has the property that 1 0 p k(x dx = p k (1 p k (0. The said polynomial is self-integrating in the closed interval [0,1]. Definition 1.1. A polynomial P in x and y with degree m, n is given by P (x, y = m,n i=0,j=0 a ij x i y j, where a ij R. Definition 1.. A function f is said to be self-integrable in the rectangular region R bounded the lines x = l 1, y = l, x = u 1 and y = u if and only if the double integral given by k k+1 u l u1 l 1 f(x, ydxdy = f(x, y x=u1 x=l 1 y=u y=l. Definition 1.3. We introduce the following terms for simplifying the double integrals and equations. Lower limit integration vector : L = (l 1, l, Upper limit integration vector : U = (l 1, l, Variable vector : V = (x, y, Integration vector : d V = dxdy. Definition 1.4. Definition 1.5. u l u1 l 1 f(x, ydxdy = U L f( V d V. f(x, y x=u1 x=l 1 y=u y=l = f( V U L.

82 76 A. C. F. Bueno No. 4. Preliminary results Lemma.1. U L f( V d V = m+n i=0,j=0 a ij (u i+1 1 l1 i+1 (u i+1 l i+1 (i + 1(j + 1, where f( V = P (x, y = m+n i=0,j=0 a ijx i y j. Lemma.. f( V U L = m+n i=1,j=1 where f( V = P (x, y = m+n i=0,j=0 a ijx i y j. a ij (u i 1 l i 1(u j lj, 3. Main results Theorem 3.1. The function f( V = P (x, y = m+n i=0,j=0 a ij x i y j is self-integrable in the rectangular region R bounded the lines x = l 1, y = l, x = u 1 and y = u if and only if the coefficients satisfy the homogeneous equation given by a 00 (u 1 l 1 (u l 1 + m+n i=0,j=0 Corollary 3.1. The function [a ij (u i+1 1 l i+1 1 (u j+1 f( V = P (x, y = l j+1 (i + 1(j + 1(u i 1 l1(u i j lj ] = 0. m+n i=0,j=0 a ij x i y j is self-integrable in the rectangular region R bounded the lines x = 0, y = 0, x = 1 and y = 1 if and only if the coefficients satisfy the homogeneous equation given by a 00 + m+n i=0,j=0 a ij [1 (i + 1(j + 1] = 0. Theorem 3.. The sum of self-integrable polynomials in the rectangular region R is also self-integrable in the region R where R is the region bounded by the lines x = l 1, y = l, x = u 1 and y = u. References [1] J. Graham, Self-Integrating Polynomials, The College Mathematics Journal, 36(005, No.4,

83 Scientia Magna Vol. 8 (01, No. 4, Signed product cordial graphs in the context of arbitrary supersubdivision P.Lawrence Rozario Raj and R. Lawrence Joseph Manoharan Department of Mathematics, St. Joseph s College Trichirappalli, 6000, Tamil Nadu, India lawraj006@yahoo.co.in Abstract In this paper we discuss signed product cordial labeling in the context of arbitrary supersubdivision of some special graphs. We prove that the graph obtained by arbitrary supersubdivision of (P n P m P s is signed product cordial. We also prove that the tadpole T n,l is signed product cordial except m i (1 i n are odd, m i (n + 1 i n + l are even and n is odd and C n P m is signed product cordial except m i (1 i n are odd, m i (n + 1 i nm are even and n is odd. Keywords Signed product cordial labeling, signed product cordial graph, tadpole, arbitrary supersubdivision. 1. Introduction and preliminaries We begin with simple, finite, connected and undirected graph G = (V (G, E(G with p vertices and q edges. For all other terminology and notations we follow Harary [3]. Given below are some definitions useful for the present investigations. Sethuraman [6] introduced a new method of construction called supersubdivision of graph. Definition 1.1. Let G be a graph with q edges. A graph H is called a supersubdivision of G if H is obtained from G by replacing every edge e i of G by a complete bipartite graph K,mi for some m i, 1 i q in such a way that the end vertices of each e i are identified with the two vertices of -vertices part of K,mi after removing the edge e i from graph G. If m i is varying arbitrarily for each edge e i then supersubdivision is called arbitrary supersubdivision of G. In the same paper Sethuraman proved that arbitrary supersubdivision of any path and cycle C n are graceful. Kathiresan [4] proved that arbitrary supersubdivision of any star are graceful. Definition 1.. The assignment of values subject to certain conditions to the vertices of a graph is known as graph labeling. Definition 1.3. Let G = (V, E be a graph. A mapping f : V (G {0, 1} is called binary vertex labeling of G and f (v is called the label of the vertex v of G under f. For an edge e = uv, the induced edge labeling f : E(G {0, 1} is given by f (e = f(u f(v.

84 78 P. Lawrence Rozario Raj and R. Lawrence Joseph Manoharan No. 4 Let v f (0, v f (1 be the number of vertices of G having labels 0 and 1 respectively under f and let e f (0, e f (1 be the number of edges having labels 0 and 1 respectively under f. Definition 1.4. A binary vertex labeling of a graph G is called a cordial labeling of G if v f (0 v f (1 1 and e f (0 e f (1 1. A graph G is cordial if it admits cordial labeling. The concept of cordial labeling was introduced by Cahit []. Vaidya [8] proved that arbitrary supersubdivision of any path, star and cycle C n except when n and all m i are simultaneously odd numbers are cordial. Vaidya [7] proved that arbitrary supersubdivision of any tree, grid graph, complete bipartite graph star and C n P m except when n and all m i are simultaneously odd numbers are cordial. The concept of signed product cordial labeling was introduced by Baskar Babujee [1]. Definition 1.5. A vertex labeling of graph G, f : V (G { 1, 1} with induced edge labeling f : E(G { 1, 1} defined by f (uv = f(uf(v is called a signed product cordial labeling if v f ( 1 v f (1 1 and e f ( 1 e f (1 1, where v f ( 1 is the number of vertices labeled with 1, v f (1 is the number of vertices labeled with 1, e f ( 1 is the number of edges labeled with 1 and e f (1 is the number of edges labeled with 1. A graph G is signed product cordial if it admits signed product cordial labeling. In the same paper Baskar Babujee proved that the path graph, cycle graph, star, and bistar are signed product cordial and some general results on signed product cordial labeling are also studied. Lawrence Rozario [5] proved that arbitrary supersubdivision of tree, complete bipartite graph, grid graph and cycle C n except when n and all m i are simultaneously odd numbers are signed product cordial. Vaidya [9] proved that arbitrary supersubdivision path, cycle, star and tadpole graph are strongly multiplicative. Definition 1.6. Tadpole T n,l is a graph in which path P l is attached by an edge to any one vertex of cycle C n. T n,l has n + l vertices and edges. Here we prove that the graphs obtained by arbitrary supersubdivision of (P n P m P s is signed product cordial. We also prove that the tadpole T n,l is signed product cordial except m i (1 i n are odd, m i (n + 1 i n + l are even and n is odd and C n P m is signed product cordial except m i (1 i n are odd, m i (n + 1 i nm are even and n is odd.. Signed product cordial labeling of arbitrary supersubdivision of graphs Theorem.1. Arbitrary supersubdivision of grid graph (P n P m P s is signed product cordial. Proof. Let v ij be the vertices of (P n P m, where 1 i n and 1 j m. Let vij k (1 i n, 1 j m and k s be the vertices of paths. Arbitrary supersubdivision of (P n P m P s is obtained by replacing every edge of (P n P m P s with K,mi and we denote the resultant graph by G.

85 Vol. 8 Signed product cordial graphs in the context of arbitrary supersubdivision 79 Let α = mn(s+1 m n 1 Let u j be the vertices which are used for arbitrary supersubdivision, where 1 j α. Here V (G = α + smn, E(G = α. We define vertex labeling f : V (G { 1, 1} as follows. f(v i,j = 1, if i and j are odd or i and j are even, 1 i n and = 1, if i is even and j is odd or i is odd and j is even. 1 j m. m i. f(vi,j k = 1, if i and j are odd or i and j are even, = 1, if i is even and j is odd or i is odd and j is even. f(vi,j k = 1, if i and j are odd or i and j are even, 1 i n, 1 j m and k is even. 1 i n, 1 j m = 1, if i is even and j is odd or i is odd and j is even. and k is odd. f(u j = 1, if j is even, = 1, if j is odd. 1 i α. In view of the above defined function f, the graph G satisfies the following conditions. When (i α and mn are even and s is odd/even, (ii α and s are even and mn is odd, (iii α and mn are odd and s is odd, then v f ( 1 = v f (1 = α + smn, e f ( 1 = e f (1 = α. When (i α and mn are odd and s is even, (ii α is odd, mn is even and s is odd/even, then v f ( = v f (1 = α + smn + 1, e f ( 1 = e f (1 = α. (iii When α is even and mn and s are odd, then v f ( 1 = v f (1 + 1 = α + smn + 1, e f ( 1 = e f (1 = α. That is, f is a signed product cordial labeling for G. Hence the result. Illustration.1 Consider the arbitrary supersubdivision of graph (P P 3 P 3. Here n =, m = 3, m 1 = 3, m =, m 3 =, m 4 =, m 5 =, m 6 =, m 7 =, m 8 =, m 9 = 3, m 10 = 1, m 11 = 1, m 1 =, m 13 =, m 14 =, m 15 =, m 16 = 1, m 17 = 1, m 18 =, m 19 = and α = 36. The signed product cordial labeling of the arbitrary supersubdivision of graph (P P 3 P 3 as shown in Figure 1.

86 80 P. Lawrence Rozario Raj and R. Lawrence Joseph Manoharan No. 4 Figure 1: Signed product cordial labeling of the arbitrary supersubdivision of (P P 3 P 3 Theorem.. Arbitrary supersubdivision of any tadpole T n,l is a signed cordial product graph except m i (1 i n are odd, m i (n + 1 i n + l are even and n is odd. Proof. Let v 1, v, v 3,..., v n be the vertices of C n and v n+1, v n+, v n+3,..., v n+l be the vertices of path P l. Arbitrary supersubdivision of T n,l is obtained by replacing every edge of T n,l with K,mi and we denote this graph by G. Let α = n+l 1 m i and u j be the vertices which are used for arbitrary supersubdivision, where 1 j α. Here V (G = α + n + l, E(G = α. To define binary vertex labeling f : V (G { 1, 1}. We consider following cases. Case 1: For n even. f(v i = 1, if i is odd, = 1, if i is even. 1 i n + l. f(u j = 1, if j is even, = 1, if j is odd. 1 j α. In view of the case, the graph G satisfies the following conditions. When (i n, l and α are even, (ii n is even, l and α are odd, then When n and l are even, α is odd, then v f ( 1 = v f (1 = n + l + α, e f ( 1 = e f (1 = α. v f ( = v f (1 = n + l + α + 1, e f ( 1 = e f (1 = α. When n and α are even, l is odd, then v f ( 1 = v f (1 + 1 = n + l + α + 1, e f ( 1 = e f (1 = α. That is, f is a signed product cordial labeling for G and consequently G is a signed product cordial graph.

87 Vol. 8 Signed product cordial graphs in the context of arbitrary supersubdivision 81 Case : For n odd. Subcase.1 : m i (1 i n are even. f(v i = 1, if i is odd, = 1, if i is even. 1 i n + l. f(u j = 1, if j is even, = 1, if j is odd. 1 j α. In view of the above case, the graph G satisfies the following conditions. When (i n and l are odd, α is even, (ii n and α are odd, l is odd, then When n is odd, l and α are even, then When n, l and α are odd, then v f ( 1 = v f (1 = n + l + α, e f ( 1 = e f (1 = α. v f ( 1 = v f (1 + 1 = n + l + α + 1, e f ( 1 = e f (1 = α. v f ( = v f (1 = n + l + α + 1, e f ( 1 = e f (1 = α. That is, f is a signed product cordial labeling for G and consequently G is a signed product cordial graph. Subcase.: At least one m i (1 i n is even. m i is even for some i, where 1 i n. In this case at least one m i must be even so label all the vertices v i+1, v i+,..., v n, v 1,..., v i alternately by using 1 and 1 starting with 1. Subcase. (a: m i is even for some i, where 1 i n and i is even or i = n. f(v i = 1, if i is odd, = 1, if i is even. n + 1 i n + l. f(u j = 1, if j is even, = 1, if j is odd. 1 j α. In view of the above case, the graph G satisfies the following conditions. When (i n and l are odd, α is even, (ii n and α are odd, l is even, then v f ( 1 = v f (1 = n+l+α, e f ( 1 = e f (1 = α. When (i n is odd, l and α are even, (ii n, l and α are odd, then v f ( 1 = v f (1 + 1 = n+l+α+1, e f ( 1 = e f (1 = α. That is, f is a signed product cordial labeling for G and consequently G is a signed product cordial graph. Subcase. (b: m i is even for some i, where 1 i < n, i is odd and l is even. f(v i = 1, if i is even, = 1, if i is odd. n + 1 i n + l. f(u j = 1, if j is even, = 1, if j is odd. 1 j α.

88 8 P. Lawrence Rozario Raj and R. Lawrence Joseph Manoharan No. 4 In view of the above case, the graph G satisfies the following conditions. When n is odd, l and α is even, then v f ( 1 = v f (1 + 1 = n + l + α + 1, e f ( 1 = e f (1 = α. When n and α are odd, l is even, then v f ( 1 = v f (1 = n + l + α, e f ( 1 = e f (1 = α. That is, f is a signed product cordial labeling for G and consequently G is a signed product cordial graph. Subcase. (c: m i is even for some i, where 1 i n, i and l are odd. f(v i = 1, if i is even, = 1, if i is odd. n + 1 i n + l. f(u j = 1, if j is even, = 1, if j is odd. 1 j α. f(u j = 1, for j = α 1, α. In view of the above two cases graph G satisfies the following conditions. When n and l are odd, α is even, then When n, l and α are odd, then v f ( 1 = v f (1 = n + l + α, e f ( 1 = e f (1 = α. v f ( = v f (1 = n + l + α + 1, e f ( 1 = e f (1 = α. That is, f is a signed product cordial labeling for G and consequently G is a signed product cordial graph. Case 3: If n is odd and m i (1 i n + l are odd. Subcase 3.1: For n odd and n + l is even. f(v i = 1, if i is odd, = 1, if i is even. 1 i n + l 1. f(v n+l = 1. f(u j = 1, if j is even, = 1, if j is odd. 1 j α m n+l. f(u j = 1, forj = α m n+l 1, α m n+l. f(u j = 1, if j is even, = 1, if j is odd. α m n+l + 1 j α. Subcase 3.: For n odd and n + l is odd. f(v i = 1, if i is odd, = 1, if i is even. 1 i n + l 1. f(v n+l = 1.

89 Vol. 8 Signed product cordial graphs in the context of arbitrary supersubdivision 83 f(u j = 1, if j is even, = 1, if j is odd. 1 j α. In view of above two cases, the graph G satisfies the following condition. v f ( 1 = v f (1 = n + l + α, e f ( 1 = e f (1 = α. That is, f is a signed product cordial labeling for G and consequently G is a signed product cordial graph. Case 4: If n is odd and m i (1 i n are odd, m i (n + 1 i n + l are even. Consider v i and v j are adjacent. If f(v i and f(v j have different values and assign any value to f(u j, then induce edge labeling of the adjacent edges have different values, i.e., if m i is either even or odd, then e f ( 1 = e f (1 = α. If f(v i and f(v j have same value and assign any value to f(u j, then induce edge labeling of the adjacent edges have same value. i.e., if m i is even, then e f ( 1 = e f (1 = α and if m i is odd, then e f ( 1 = e f (1 + or e f ( 1 + = e f (1. Here n is odd and all m i (1 i n are odd in cycle C n. Then there exist atleast one pair of adjacent vertices v i and v j of cycle C n have f(v i and f(v j are same. Therefore, assign any value to f(u j, then induce edge labeling of the adjacent edges have same value and e f ( 1 = e f (1 + or e f ( 1 + = e f (1. But m i (n + 1 i n + l are even. Therefore, assign any value to f(u j, then induce edge labeling of the adjacent edges have different values. Here n is odd, m i (1 i n are odd and m i (n + 1 i n + l are even, then either e f ( 1 = e f (1 + or e f ( 1 + = e f (1. Therefore, G is not signed product cordial. Illustration. Consider the arbitrary supersubdivision of T 4,3. Here n = 4, l = 3, m 1 =, m =, m 3 =, m 4 =, m 5 = 1, m 6 =, m 7 = 3 and α = 14. The signed product cordial labeling of the arbitrary supersubdivision of T 4,3 as shown in Figure. Figure : Signed product cordial labeling of the arbitrary supersubdivision of T 4,3 Theorem.3: Arbitrary supersubdivision of C n P m is signed product cordial except m i (1 i n are odd, m i (n + 1 i nm are even and n is odd. Proof. Let v 1, v, v 3,..., v n be the vertices of C n and v ij (1 i n, j m be the vertices of paths. Arbitrary supersubdivision of C n P m is obtained by replacing every edge of C n P m with K,mi and we denote this graph by G. Let α = mn 1 m i and u j be the vertices which are used for arbitrary supersubdivision, where 1 j α. Here V (G = α + mn, E(G = α. To define binary vertex labeling f : V (G { 1, 1}. We consider following cases.

90 84 P. Lawrence Rozario Raj and R. Lawrence Joseph Manoharan No. 4 Case 1: For n even. f(v i = 1, if i is odd, = 1, if i is even, 1 i n and f(v ij = 1, if i and j are odd or i and j are even, j m. = 1, if i is even and j is odd or i is odd and j is even. f(u j = 1, if j is even, = 1, if j is odd. 1 j α. In view of the above case, the graph G satisfies the following conditions. When α + mn is even, then v f ( 1 = v f (1 = α + mn, e f ( 1 = e f (1 = α. When α is odd and mn is even, then v f ( = v f (1 = α + mn + 1, e f ( 1 = e f (1 = α. That is, f is a signed product cordial labeling for G and consequently G is a signed product cordial graph. Case : For n odd and at least one m i (1 i n is even. Without loss of generality we assume that m 1 is even. f(v 1 = 1. f(v i = 1, if i is even, = 1, if i is odd, f(v 1j = 1, if j is odd, = 1, if j is even, i n and j m. f(v ij = 1, if i is even and j is odd or i is odd and j is even, = 1, if i and j are odd or i and j are even. f(u j = 1, if 1 j m1, = 1, if m1 + 1 j m 1, 1 j α. f(u j = 1, if m j α+m1, = 1, if α+m1 j α. In view of the above case, the graph G satisfies the following conditions. When α + mn is even, then v f ( 1 = v f (1 = α + mn, e f ( 1 = e f (1 = α. When α is odd and mn is even, then When α is even and mn is odd, then v f ( = v f (1 = α + mn + 1, e f ( 1 = e f (1 = α. v f ( 1 = v f (1 + 1 = α + mn + 1, e f ( 1 = e f (1 = α.

91 Vol. 8 Signed product cordial graphs in the context of arbitrary supersubdivision 85 That is, f is a signed product cordial labeling for G and consequently G is a signed product cordial graph. Case 3: If n is odd and m i (1 i nm are odd. Subcase 3.1: For n odd and nm is even. f(u j = 1, if j is odd, = 1, if j is even. 1 j m 1. f(u j = 1, if m j m 1 + α mnm m1, = 1, if m 1 + α mnm m1 + 1 j α m nm, f(u j = 1, if α m nm + 1 j α m nm + mnm, = 1, if α m nm + mnm + 1 j α. f(v 1 = 1, f(v i = 1, if i is even, = 1, if i is odd, f(v 1j = 1, if j is odd, = 1, if j is even, f(v ij = 1, if i is even and j is odd or i is odd f(v nm = 1. and j is even and i n and j m, = 1, if i and j are odd or i and j are even, 1 j α. i n and j m. In view of above, the graph G satisfies the following condition. Here α + mn is even, then v f ( 1 = v f (1 = α + mn, e f ( 1 = e f (1 = α. That is, f is a signed product cordial labeling for G and consequently G is a signed product cordial graph. Subcase 3.: For n odd and nm is odd. Here β = m 1 +m +...+m n +m n m n+(m, f(v 1 = 1, f(v i = 1, if i is even, = 1, if i is odd, f(v 1j = 1, if j is odd and j m, = 1, if j is even, f(v ij = 1, if i is even and j is odd or i is odd and j is even, f(v 1m = 1. = 1, if i and j are odd or i and j are even, i n and j m.

92 86 P. Lawrence Rozario Raj and R. Lawrence Joseph Manoharan No. 4 f(u j = 1, if 1 j m1, = 1, if m1 + 1 j m 1, f(u j = 1, if m j m 1 + β m1, = 1, if m 1 + β m1 + 1 j β, f(u j = 1, if β + 1 j β + m n+(m 1, = 1, if β + m n+(m j β + m n+(m 1, f(u j = 1, if β + m n+(m j β + m n+(m 1 1 j α. + α β m n+(m 1, = 1, if β + m n+(m 1 + α β m n+(m j α. In view of above, the graph G satisfies the following condition. Here α + mn is even, then v f ( 1 = v f (1 = α + mn, e f ( 1 = e f (1 = α. i.e., f is a signed product cordial labeling for G and consequently G is a signed product cordial graph. Case 4: If n is odd and m i (1 i n are odd, m i (n + 1 i nm are even. Here n is odd and all m i (1 i n are odd in cycle C n. Then there exist at least one pair of adjacent vertices v i and v j of cycle C n have f(v i and f(v j are same. Therefore, assign any value to f(u j, then induce edge labeling of the adjacent edges have same value and e f ( 1 = e f (1 + or e f ( 1 + = e f (1. But m i (n + 1 i nm are even. Therefore, assign any value to f(u j, then induce edge labeling of the adjacent edges have different values. Here n is odd, m i (1 i n are odd and m i (n + 1 i nm are even, then either e f ( 1 = e f (1 + or e f ( 1 + = e f (1. Therefore, G is not signed product cordial. Illustration.3 : Consider the arbitrary supersubdivision of C 4 P 3. Here n = 4, m = 3, m 1 =, m =, m 3 =, m 4 =, m 5 = 1, m 6 = 1, m 7 =, m 8 =, m 9 =, m 10 =, m 11 = 1, m 1 = 1 and α = 0. The signed product cordial labeling of the arbitrary supersubdivision of graph C 4 P 3 as shown in Figure 3. Figure 3: Signed product cordial labeling of the arbitrary supersubdivision of C 4 P 3

93 Vol. 8 Signed product cordial graphs in the context of arbitrary supersubdivision 87 References [1] J. Baskar Babujee, Shobana Loganathan, On Signed Product Cordial Labeling, Applied Mathematics, (011, [] I. Cahit, Cordial Graphs: A weaker version of graceful and harmonious Graphs, Ars Combinatoria, 3(1987, [3] F. Harary, Graph theory, Addison Wesley, Reading, Massachusetts, 197. [4] K. M. Kathiresan, S. Amutha, Arbitrary supersubdivisions of stars are graceful, Indian J. pure appl. Math., 35(004, No. 1, [5] P. Lawrence Rozario Raj, R. Lawrence Joseph Manoharan, Signed Product Cordialness of Arbitrary Supersubdivisions of Graphs, International Journal of Mathematics and Computational Methods in Science and Technology, (01, No., 1-6. [6] G. Sethuraman, P. Selvaraju, Gracefulness of arbitrary supersubdivisions of graphs, Indian J. pure appl. Math., 3 (001, No. 7, [7] S. K. Vaidya, N. A. Dani, Cordial labeling and arbitrary super subdivision of some graphs, International Journal of Information Science and Computer Mathematics, (010, No. 1, [8] S. K. Vaidya, K. K. Kanani, Some new results on cordial labeling in the context of arbitrary supersubdivision of graph, Appl. Math. Sci., 4 (010, No. 47, [9] S. K. Vaidya, K. K. Kanani, Some strongly multiplicative graphs in the context of arbitrary supersubdivision, International Journal of Applied Mathematics and Computation, 3(011, No. 1,

94 Scientia Magna Vol. 8 (01, No. 4, Smarandache cyclic geometric determinant sequences A. C. F. Bueno Department of Mathematics and Physics, Central Luzon State University, Science City of Muñoz 310, Nueva Ecija, Philippines Abstract In this paper, the concept of Smarandache cyclic geometric determinant sequence was introduced and a formula for its n th term was obtained using the concept of right and left circulant matrices. Keywords Smarandache cyclic geometric determinant sequence, determinant, right circulan -t matrix, left circulant matrix. 1. Introduction and preliminaries Majumdar [1] gave the formula for n th term of the following sequences: Smarandache cyclic natural determinant sequence, Smarandache cyclic arithmetic determinant sequence, Smarandache bisymmetric natural determinant sequence and Smarandache bisymmetric arithmetic determinant sequence. Definition 1.1. A Smarandache cyclic geometric determinant sequence {SCGDS(n} is a sequence of the form a {SCGDS(n} = a, ar a ar ar ar a, ar ar a,.... ar a ar Definition 1.. A matrix RCIRC n ( c M nxn (R is said to be a right circulant matrix if it is of the form c 0 c 1 c... c n c n 1 c n 1 c 0 c 1... c n 3 c n c n c n 1 c 0... c n 4 c n 3 RCIRC n ( c =., c c 3 c 4... c 0 c 1 c 1 c c 3... c n 1 c 0 where c = (c 0, c 1, c,..., c n, c n 1 is the circulant vector.

95 Vol. 8 Smarandache cyclic geometric determinant sequences 89 Definition 1.3. A matrix LCIRC n ( c M nxn (R is said to be a leftt circulant matrix if it is of the form c 0 c 1 c... c n c n 1 c 1 c c 3... c n 1 c 0 c c 3 c 4... c 0 c 1 LCIRC n ( c =., c n c n 1 c 0... c n 4 c n 3 c n 1 c 0 c 1... c n 4 c n where c = (c 0, c 1, c,..., c n, c n 1 is the circulant vector. Definition 1.4. A right circulant matrix RCIRC n ( g with geometric sequence is a matrix of the form a ar ar... ar n ar n 1 ar n 1 a ar... ar n 3 ar n ar n ar n 1 a... ar n 4 ar n 3 RCIRC n ( g = ar ar 3 ar 4... a ar ar ar ar 3... ar n 1 a. Definition 1.5. A left circulant matrix LCIRC n ( g with geometric sequence is a matrix of the form a ar ar... ar n ar n 1 ar ar ar 3... ar n 1 a ar ar 3 ar 4... a ar LCIRC n ( g = ar n ar n 1 a... ar n 4 ar n 3 ar n 1 a ar... ar n 4 ar n. The right and left circulant matrices has the following relationship: where Π = O = O T 1. 1 O 1 O Ĩ n 1 LCIRC n ( c = ΠRCIRC n ( c with Ĩn 1 = , O 1 = ( and

96 90 A. C. F. Bueno No. 4 Clearly, the terms of {SCGDS(n} are just the determinants of LCIRC n ( g. Now, for the rest of this paper, let A be the notation for the determinant of a matrix A. Hence {SCGDS(n} = { LCIRC 1 ( g, LCIRC ( g, LCIRC 3 ( g,...}.. Preliminary results Lemma.1. RCIRC n ( g = a n (1 r n n 1. Proof. RCIRC n ( g = = a a ar ar... ar n ar n 1 ar n 1 a ar... ar n 3 ar n ar n ar n 1 a... ar n 4 ar n ar ar 3 ar 4... a ar ar ar ar 3... ar n 1 a 1 r r... r n r n 1 r n 1 1 r... r n 3 r n r n r n r n 4 r n r r 3 r r r r r 3... r n 1 1. By applying the row operations r n k R 1 + R k+1 R k+1 where k = 1,, 3,..., n 1, 1 r r... r n r n 1 0 (r n 1 r(r n 1... r n 3 (r n 1 r n (r n (r n 1... r n 4 (r n 1 r n 3 (r n 1 RCIRC n ( g a (r n 1 r(r n (r n 1 Since ca = c n A and its row equivalent matrix is a lower traingular matrix it follows that RCIRC n ( g = a n (1 r n n 1. Lemma.. Π = ( 1 n 1, where x is the floor function. Proof. Case 1: n = 1,, Π = I n = 1.

97 Vol. 8 Smarandache cyclic geometric determinant sequences 91 Case : n is even and n > If n is even then there will be n rows to be inverted because there are two 1 s in the main diagonal. Hence there will be n inversions to bring back Π to I n so it follows that Π = ( 1 n. Case 3: n is odd and and n > If n is odd then there will be n 1 rows to be inverted because of the 1 in the main diagonal of the frist row. Hence there will be n 1 inversions to bring back Π to I n so it follows that But n 1 = n, so the lemma follows. 3. Main results Π = ( 1 n 1. Theorem 3.1. The n th term of {SCGDS(n} is given by via the previous lemmas. SCGDS(n = ( 1 n 1 a n (1 r n n 1 References [1] M. Bahsi and S. Solak, On the Circulant Matrices with Arithmetic Sequence, Int. J. Contemp. Math. Sciences, 5, 010, No. 5, [] H. Karner, J. Schneid, C. Ueberhuber, Spectral decomposition of real circulant matrices, Institute for Applied Mathematics and Numerical Analysis, Vienna University of Technology, Austria. [3] A. A. K. Majumdar, On some Smarandache determinant sequence, Scientia Magna, 4 (008, No.,

98 Scientia Magna Vol. 8 (01, No. 4, convergence results for asymptotically generalized Φ-hemicontractive mappings A. A. Mogbademu Abstract Department of Mathematics University of Lagos, Nigeria amogbademu@unilag.edu.ng prinsmo@yahoo.com In this paper, some strong convergence results are obtained for asymp totically generalized Φ-hemicontractive maps in real Banach spaces using a modified Mann iteration formula with errors. Our results extend and improve the convergence results of Kim, Sahu and Nam [7]. Keywords Modified mann iteration, noor iteration, uniformly lipschitzian, asymptotically p -seudocontractive, three-step iterative scheme, Banach spaces. 010 Mathematics Subject Classsification: 47H10, 46A03 1. Introduction and preliminaries We denote by J the normalized duality mapping from X into X by J(x = {f X : x, f = x = f }, where X denotes the dual space of real normed linear space X and.,. denotes the generalized duality pairing between elements of X and X. We first recall and define some concepts as follows (see, [7]. Let C be a nonempty subset of real normed linear space X. Definition 1.1. A mapping T : C X is called strongly pseudocontractive if for all x, y C, there exist j(x y J(x y and a constant k (0, 1 such that T x T y, j(x y (1 k x y. A mapping T is called strongly φ-pseudocontractive if for all x, y C, there exist j(x y J(x y and a strictly increasing function φ : [0, [0, with φ(0 = 0 such that T x T y, j(x y x y φ( x y x y and is called generalized strongly Φ-pseudocontractive if for all x, y C, there exist j(x y J(x y and a strictly increasing function Φ : [0, [0, with Φ(0 = 0 such that T x T y, j(x y x y Φ( x y.

99 Vol. 8 Some convergence results for asymptotically generalized Φ-hemicontractive mappings 93 Every strongly φ-pseudocontractive operator is a generalized strongly Φ-pseudo contractive operator with Φ : [0, [0, defined by Φ(s = φ(ss, and every strongly pseudocontractive operator is strongly φ-pseudocontractive operator where φ is defined by φ(s = ks for k (0, 1 while the converses need not be true. An example by Hirano and Huang [6] showed that a strongly φ-pseudocontractive operator T is not always a strongly pseudocontractive operator. A mapping T is called generalized Φ-hemicontractive if F (T = {x C : x = T x} = and for all x C and x F (T, there exist j(x x J(x x and a strictly increasing function Φ : [0, [0, with Φ(0 = 0 such that T x T x, j(x x x x Φ( x x. Definition 1.. A mapping T : C C is called asymptotically generalized Φ-pseudocontractive with sequence {k n } if for each n N and x, y C, there exist constant k n 1 with lim n k n = 1, strictly increasing function Φ : [0, [0, with Φ(0 = 0 such that T n x T n y, j(x y k n x y Φ( x y, and is called asymptotically generalized Φ-hemicontractive with sequence {k n } if F (T and for each n N and x C, x F (T, there exist constant k n 1 with lim n k n = 1, strictly increasing function Φ : [0, [0, with Φ(0 = 0 such that T n x T n x, j(x x k n x x Φ( x x. Clearly, the class of asymptotically generalized Φ-hemicontractive mappings is the most general among those defined above. (see, []. Definition 1.3. A mapping T : C X is called Lipschitzian if there exists a constant L > 0 such that T x T y L x y for all x, y C and is called generalized Lipschitzian if there exists a constant L > 0 such that for all x, y C. T x T y L( x y + 1 A mapping T : C C is called uniformly L-Lipschitzian if for each n N, there exists a constant L > 0 such that for all x, y C. T n x T n y L x y It is obvious that the class of generalized Lipschitzian map includes the class of Lipschitz map. Moreover, every mapping with a bounded range is a generalized Lipschitzian mapping. Sahu [6] introduced the following new class of nonlinear mappings which is more general than the class of generalized Lipschitzian mappings and the class of uniformly L-Lipschitzian mappings. Fix a sequence {r n } in [0, ] with r n 0.

100 94 A. A. Mogbademu No. 4 Definition 1.4. A mapping T : C C is called nearly Lipschitzian with respect to {r n } if for each n N, there exists a constant k n > 0 such that T n x T n y k n ( x y + r n for all x, y C. A nearly Lipschitzian mapping T with sequence {r n } is said to be nearly uniformly L- Lipschitzian if k n = L for all n N. Observe that the class of nearly uniformly L-Lipschitzian mapping is more general than the class of uniformly L-Lipschitzian mappings. In recent years, many authors have given much attention to iterative methods for approximating fixed points of Lipschitz type pseudocontractive type nonlinear mappings (see, [1-4, 7, 9-14]. Ofoedu [9] used the modified Mann iteration process introduced by Schu [1] x n+1 = (1 α n x n + α n T n x n, n 0 (1 to obtain a strong convergence theorem for uniformly Lipschitzian asymptotically pseudocontractive mapping in real Banach space setting. This result itself is a generalization of many of the previous results (see [9] and the references therein. Recently, Chang [3] proved a strong convergence theorem for a pair of L-Lipschitzian mappings instead of a single map used in [9]. In fact, they proved the following theorem. Theorem 1.1. [3] Let E be a real Banach space, K be a nonempty closed convex subset of E, T i : K K, (i = 1, be two uniformly L i -Lipschitzian mappings with F (T 1 F (T φ, where F (T i is the set of fixed points of T i in K and ρ be a point in F (T 1 F (T. Let k n [1, be a sequence with k n 1. Let {α n } n=1 and {β n } n=1 be two sequences in [0, 1] satisfying the following conditions: (i n=1 α n =, (ii n=1 α n <, (iii n=1 β n <, (iv n=1 α n(k n 1 <. For any x 1 K, let {x n } n=1 be the iterative sequence defined by x n+1 = (1 α n x n + α n T n 1 y n. y n = (1 β n x n + β n T n x n. ( If there exists a strictly increasing function Φ : [0, [0, with Φ(0 = 0 such that < T n 1 x n ρ, j(x n ρ > k n x n ρ Φ( x n ρ for all j(x ρ J(x ρ and x K, (i = 1,, then {x n } n=1 converges strongly to ρ. The result above extends and improves the corresponding results of [9] from one uniformly Lipschitzian asymptotically pseudocontractive mapping to two uniformly Lipschitzian asymptotically pseudocontractive mappings. In fact, if the iteration parameter {β n } n=0 in Theorem 1.1 above is equal to zero for all n and T 1 = T = T then, we have the main result of Ofoedu [9].

101 Vol. 8 Some convergence results for asymptotically generalized Φ-hemicontractive mappings 95 Very recently, Kim, Sahu and Nam [7] used the notion of nearly uniformly L-Lipschitzian to established a strong convergence result for asymptotically generalized Φ-hemicontractive mappings in a general Banach space. This result itself is a generalization of many of the previous results. (see, [7]. Indeed, they proved the following: Theorem 1.. [7] Let C be a nonempty closed convex subset of a real Banach space X and T : C C a nearly uniformly L-Lipschitzian mappings with sequence {r n } and asymptotically generalized Φ hemicontractive mapping with sequence {k n } and F (T. Let {α n } be a sequence in [0, 1] satisfying the conditions: (i { rn α n } is bounded, (ii n=1 α n =, (iii n=1 α n < and n=1 α n(k n 1 <. Then the sequence {x n } n=1 in C defined by (1 converges to a unique fixed point of T. Remark 1.1. Although Theorem 1. is a generalization of many of the previous results but, we observed that, the proof process of Theorem 1. in [7] can be extend and improve upon. In this paper, we employed a simple analytical approach to prove the convergence of the modified Mann iteration with errors for fixed points of uniformly L-Lipschitzian asymptotically generalized Φ-hemicontractive mapping and improve the results of Kim, Sahu and Nam [7]. For this, we need the following Lemmas. Lemma 1.1. [3] Let X be a real Banach space and let J : X x be a normalized duality mapping. For all x, y X and for all j(x + y J(x + y. Then x + y x + < y, j(x + y >. Lemma 1.. [8] Let Φ : [0, [0, be an increasing function with Φ(x = 0 x = 0 and let {b n } n=0 be a positive real sequence satisfying b n = + and n=0 lim b n = 0. n Suppose that {a n } n=0 is a nonnegative real sequence. If there exists an integer N 0 > 0 satisfying a n+1 < a n + o(b n b n Φ(a n+1, n N 0, where then lim n a n = 0. o(b n lim = 0, n b n. Main results Theorem.1. Let C be a nonempty closed convex subset of a real Banach space X and T : C C a nearly uniformly L-Lipschitzian mapping with sequence {r n }. Let {a n } n=1, {b n } n=1 and {c n } n=1 be real sequences in [0, 1] satisfying (i a n + b n + c n = 1, (ii 1 b n is bounded,

102 96 A. A. Mogbademu No. 4 (iii n 0 b n =, (iv c n = o(b n, (v lim n b n = 0. Let T be asymptotically generalized Φ-hemicontractive mapping with sequence k n [1,, k n 1 such that ρ F (T = {x C : T x = x} and {x n } n=1 be the sequence generated for x 1 C by x n+1 = a n x n + b n T n x n + c n u n, n 1. (3 Then the sequence {x n } n=1 in C defined by (3 converges to a unique fixed point of T. Proof. Since T is a nearly uniformly L-Lipschitzian, asymptotically generalized Φ-hemicontractive mapping, then there exists a strictly increasing continuous function Φ : [0, [0, with Φ(0 = 0 such that T n x n T n ρ L( x n ρ + r n and T n x n T n ρ, j(x n ρ k n x n ρ Φ( x n ρ (4 for x C, ρ F (T. Equation (3 reduces to x n+1 = (1 b n x n + b n T n x n + c n (u n x n, n 1. (5 Step 1. We first show that {x n } n=1 is a bounded sequence. For this, if x n0 = T x n0, n 1 then it clearly holds. So, let if possible, there exists a positive integer x n0 C such that x n0 T x n0, thus set x n0 = x 0 and a 0 = x 0 T n x 0 x 0 ρ + (k n 1 x 0 ρ. Thus by (4, so that, on simplifying T n x 0 T n ρ, j(x ρ k n x 0 ρ Φ( x 0 ρ, (6 x 0 ρ Φ 1 (a 0. (7 Now, we claim that x n ρ Φ 1 (a 0, n 0. Clearly, inview of (7, the claim holds for n = 0. We next assume that x n ρ Φ 1 (a 0, for some n and we shall prove that x n+1 ρ Φ 1 (a 0. Suppose this is not true, i.e. x n+1 ρ > Φ 1 (a 0. 1 Since {r n } [0, ] with r n 0, b n and {u n } are bounded sequences, set M 1 = sup{r n : n N}, M 3 = sup{ 1 b n : n N} and M = sup{u n : n N}. Denote τ 0 = min 1 3 {1, Φ((Φ 1 (a 0 18(Φ 1 (a 0, Φ((Φ 1 (a 0 6(1 + L[( + LΦ 1 (a 0 + M 1 L + M + M1M3L (1+L ](Φ 1 (a 0, Φ((Φ 1 (a 0 6Φ 1 (a 0 (M + Φ 1 (a 0, 3Φ 1 (a 0 ( + LΦ 1 }. (8 (a 0 + M 1 L + M Since lim n b n, c n = 0, and c n = o(b n, without loss of generality, let 0 b n, c n, k n 1 τ 0, c n < b n τ 0 for any n 1. Then we have the following estimates from (3.

103 Vol. 8 Some convergence results for asymptotically generalized Φ-hemicontractive mappings 97 x n T n x n x n ρ + T n x n ρ x n ρ + L( x n ρ + r n (1 + L x n ρ + r n L (1 + LΦ 1 (a 0 + M 1 L. u n x n = (u n ρ + ρ x n u n ρ + x n ρ M + Φ 1 (a 0. x n+1 ρ (1 b n x n + b n T n x n + c n (u n x n ρ, x n ρ + b n T n x n x n + c n u n x n, Φ 1 (a 0 + b n [(1 + LΦ 1 (a 0 + M 1 L] +c n [M + Φ 1 (a 0 ] Φ 1 (a 0 + τ 0 [( + LΦ 1 (a 0 + M 1 L + M ] 3Φ 1 (a 0. x n x n+1 τ 0 [( + LΦ 1 (a 0 + M 1 L + M ]. = b n (T n x n x n + c n (u n x n b n T n x n x n + c n u n x n b n [(1 + LΦ 1 (a 0 + M 1 L] + c n [M + Φ 1 (a 0 ] τ 0 [( + LΦ 1 (a 0 + M 1 L + M ]. (9 Using Lemma 1.1 and the above estimates, we have x n+1 ρ x n ρ b n < x n T n x n, j(x n+1 ρ > +c n < u n x n, j(x n+1 ρ > = x n ρ + b n < T n x n+1 ρ, j(x n+1 ρ > b n < x n+1 ρ, j(x n+1 ρ > +b n < T n x n T n x n+1, j(x n+1 ρ > +b n < x n+1 x n, j(x n+1 ρ > +c n < u n x n, j(x n+1 ρ > x n ρ + b n (k n x n+1 ρ Φ( x n+1 ρ b n x n+1 ρ + b n L( x n+1 x n + r n x n+1 ρ +b n x n+1 x n x n+1 ρ + c n u n x n x n+1 ρ x n ρ + b n (k n 1 x n+1 ρ b n Φ( x n+1 ρ +b n {(1 + L x n+1 x n + M 1 L} x n+1 ρ +c n (M + x n ρ x n+1 ρ x n ρ b n Φ((Φ 1 (a b n (k n 1(Φ 1 (a 0 +6b n Φ 1 (a 0 {(1 + Lτ 0 (( + LΦ 1 (a 0 +M 1 L + M + M 1 L} + 6b n Φ 1 (a 0 (M + Φ 1 (a 0 τ 0 = x n ρ b n Φ((Φ 1 (a b n τ 0 (Φ 1 (a 0

104 98 A. A. Mogbademu No. 4 +6b n Φ 1 (a 0 (1 + Lτ 0 {( + LΦ 1 (a 0 +M 1 L + M + M 1L (1 + Lτ 0 } + 6b n Φ 1 (a 0 (M + Φ 1 (a 0 τ 0, x n ρ b n Φ((Φ 1 (a b n τ 0 (Φ 1 (a 0 +6b n Φ 1 (a 0 (1 + Lτ 0 {( + LΦ 1 (a 0 +M 1 L + M + M 1L (1 + Lb n } + 6b n Φ 1 (a 0 (M + Φ 1 (a 0 τ 0, = x n ρ b n Φ((Φ 1 (a b n τ 0 (Φ 1 (a 0 +6b n Φ 1 (a 0 (1 + Lτ 0 {( + LΦ 1 (a 0 +M 1 ra x n ρ b n Φ((Φ 1 (a 0 x n ρ ((Φ 1 (a 0, (10 which is a contradiction. Hence {x n } n=1 is a bounded sequence. Step. We want to prove x n ρ. Since b n, c n 0 as n and {x n } n=1 is bounded. From (9, we observed that lim n x n+1 x n = 0, lim n T n x n T n x n+1 = 0, lim n (k n 1 = 0. So from (10, we have x n+1 ρ x n ρ b n < x n T n x n, j(x n+1 ρ > +c n < u n x n, j(x n+1 ρ > = x n ρ + b n < T n x n+1 ρ, j(x n+1 ρ > b n < x n+1 ρ, j(x n+1 ρ > +b n < T n x n T n x n+1, j(x n+1 ρ > +b n < x n+1 x n, j(x n+1 ρ > +c n < u n x n, j(x n+1 ρ > x n ρ + b n (k n 1 x n+1 ρ b n Φ( x n+1 ρ b n x n+1 x n x n+1 ρ +c n u n x n x n+1 ρ = x n ρ b n Φ( x n+1 ρ + o(b n, where b n (k n 1 x n+1 ρ + b n x n+1 x n x n+1 ρ +b n T n x n+1 T n x n x n+1 ρ + c n u n x n x n+1 ρ = o(b n. (11 By Lemma 1., we obtain that This completes the proof. lim x n ρ = 0. n

105 Vol. 8 Some convergence results for asymptotically generalized Φ-hemicontractive mappings 99 Corollary.1. Let C be a nonempty closed convex subset of a real Banach space X and T : C C a nearly uniformly L-Lipschitzian mapping with {r n }. Let {α n } n=1 be a real sequence in [0, 1] satisfying: (i 1 α n is bounded, (ii n 0 α n =, (iii lim n α n = 0. Let T be asymptotically generalized Φ hemicontractive mapping with sequence k n [1,, k n 1 such that ρ F (T = {x C : T x = x} and {x n } n=1 be the sequence generated for x 1 C by x n+1 = (1 α n x n + α n T n x n, n 1, (1 Then the sequence {x n } n=1 in C defined by (1 converges to a unique fixed point of T. Remark.1. Our Corollary removes the conditions in Theorem.1 of [7], (i.e. n=1 α n <, n=1 α n(k n 1 < by replacing them with a weaker condition lim n α n = 0. Therefore our result extends and improves the very recent results of Kim [7] which in turn is a correction, improvement and generalization of several results. Application.1. Let X = R, C = [0, 1] and T : C C be a map defined by T x = x 4. Clearly, T is nearly uniformly Lipschitzian (r n = 1 4 n with F (T = 0. Define Φ : [0, + [0, + by Φ(t = t 4, then Φ is a strictly increasing function with Φ(0 = 0. For all x C, ρ F (T, we get < T n x T n ρ, j(x ρ > = < xn 0, j(x 0 > 4n = < xn 4 n 0, x > = xn+1 4 n x x 4 x Φ(x. Obviously, T is asymptotically generalized Φ-hemicontractive mapping with sequence {k n } = 1. If we take b n = 1 n + 1, c 1 n = (n + 1 for all n 1. For arbitrary x 1 C, the sequence {x n } n=1 C defined by (3 converges strongly to the unique fixed point ρ T.

106 100 A. A. Mogbademu No. 4 References [1] S. S. Chang, Some results for asymptotically pseudocontractive mappings and asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 19(000, [] S. S. Chang, Y. J. Cho, B. S. Lee and S. H. Kang, Iterative approximation of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces, J. Math. Anal. Appl., 4(1998, [3] S. S. Chang, Y. J. Cho, J. K. Kim, Some results for uniformly L-Lipschitzian mappings in Banach spaces, Applied Mathematics Letters, (009, [4] K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proceedings of American Mathematical Society, vol. 35(197, [5] N. Hirano and Z. Huang, Convergence theorems for multivalue φ-hemicontractive operators and φ-strongly accretive operators, Comput. Math. Appl., 36(1998, No., [6] Z. Huang, Equivalence theorems of the convergence between Ishikawa and Mann iterations with errors for generalized strongly successively Φ-pseudocontractive mappings without Lipschitzian assumptions, J. Math. Anal. Appl., 39(007, [7] J. K. Kim, D. R. Sahu and Y. M. Nam, Convergence theorem for fixed points of nearly uniformly L-Lipschitzian asymptotically generalized Φ-hemicontractive mappings, Nonlinear Analysis 71(009, [8] C. Moore and B. V. C. Nnoli, Iterative solution of nonlinear equations involving setvalued uniformly accretive operators, Comput. Math. Anal. Appl., 4(001, [9] E. U. Ofoedu, Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space, J. Math. Anal. Appl., 31(006, [10] J. O. Olaleru and A. A. Mogbademu, Modified Noor iterative procedure for uniformly continuous mappings in Banach spaces, Boletin de la Asociacion Matematica Venezolana, XVIII(011, No., [11] A. Rafiq, A. M. Acu and F. Sofonea, An iterative algorithm for two asymptotically pseudocontractive mappings, Int. J. Open Problems Compt. Math., (009, [1] J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl., 158(1999, [13] D. R. Sahu, Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces, Comment. Math. Univ. Carolin, 46(005, No.4, [14] X. Xue, The equivalence among the modified Mann-Ishikawa and Noor iterations for uniformly L-Lipschitzian mappings in Banach spaces. Journal Mathematical Inequalities, (reprint(008, 1-10.

107 Scientia Magna Vol. 8 (01, No. 4, The v invariant χ sequence spaces Nagarajan Subramanian and Umakanta Misra Abstract ( χ. Department of Mathematics, Sastra University, Thanjavur , India Department of Mathematics and Statistics, Berhampur University, Berhampur, ,Odissa, India nsmaths@yahoo.com umakanta misra@yahoo.com In this paper we characterize v-invariance of the sequence spaces ( Λ, and Keywords Gai sequence, analytic sequence, modulus function, double sequences. 010 Mathematics Subject Classification: 40A05, 40C05, 40D Introduction Throughout w, χ and Λ denote the classes of all, gai and analytic scalar valued single sequences, respectively. We write w for the set of all complex sequences (x mn, where m, n N, the set of positive integers. Then, w is a linear space under the coordinate wise addition and scalar multiplication. Some initial work on double sequence spaces is found in Bromwich [4]. Later on, they were investigated by Hardy [5], Moricz [9], Moricz and Rhoades [10], Basarir and Solankan [], Tripathy [17], Turkmenoglu [19], and many others. Let us define the following sets of double sequences: { } M u (t := (x mn w : sup m,n N x mn tmn <, C p (t := { } (x mn w : p lim m,n x mn l tmn = 1 for some l C, C 0p (t := L u (t := { } (x mn w : p lim m,n x mn tmn = 1, { (x mn w : } m=1 n=1 x mn tmn <, C bp (t := C p (t M u (t and C 0bp (t = C 0p (t M u (t; where t = (t mn is the sequence of strictly positive reals t mn for all m, n N and p lim m,n denotes the limit in the Pringsheim s sense. In the case t mn = 1 for all m, n N, M u (t, C p (t, C 0p (t, L u (t, C bp (t and C 0bp (t reduce to the sets M u, C p, C 0p, L u, C bp and C 0bp, respectively. Now, we may summarize the knowledge given in some document related to the double sequence spaces. Gökhan and Colak [1,] have proved that M u (t and

108 10 Nagarajan Subramanian and Umakanta Misra No. 4 C p (t, C bp (t are complete paranormed spaces of double sequences and gave the α-, β-, γ-duals of the spaces M u (t and C bp (t. Quite recently, in her PhD thesis, Zelter [3] has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely [4] have recently introduced the statistical convergence and Cauchy for double sequences and given the relation between statistical convergent and strongly Cesàro summable double sequences. Nextly, Mursaleen [5] and Mursaleen and Edely [6] have defined the almost strong regularity of matrices for double sequences and applied these matrices to establish a core theorem and introduced the M-core for double sequences and determined those four dimensional matrices transforming every bounded double sequences x = (x jk into one whose core is a subset of the M-core of x. More recently, Altay and Basar [7] have defined the spaces BS, BS (t, CS p, CS bp, CS r and BV of double sequences consisting of all double series whose sequence of partial sums are in the spaces M u, M u (t, C p, C bp, C r and L u, respectively, and also have examined some properties of those sequence spaces and determined the α-duals of the spaces BS, BV, CS bp and the β (ϑ-duals of the spaces CS bp and CS r of double series. Quite recently Basar and Sever [8] have introduced the Banach space L q of double sequences corresponding to the well-known space l q of single sequences and have examined some properties of the space L q. Quite recently Subramanian and Misra [9] have studied the space χ M (p, q, u of double sequences and have given some inclusion relations. Spaces are strongly summable sequences was discussed by Kuttner [31], Maddox [3], and others. The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox [8] as an extension of the definition of strongly Cesàro summable sequences. Connor [33] further extended this definition to a definition of strong A-summability with respect to a modulus where A = (a n,k is a nonnegative regular matrix and established some connections between strong A-summability, strong A-summability with respect to a modulus, and A-statistical convergence. In [34] the notion of convergence of double sequences was presented by A. Pringsheim. Also, in [35]-[38], and [39] the four dimensional matrix transformation (Ax k,l = m=1 n=1 amn kl x mn was studied extensively by Robison and Hamilton. In their work and throughout this paper, the four dimensional matrices and double sequences have real-valued entries unless specified otherwise. In this paper we extend a few results known in the literature for ordinary (single sequence spaces to multiply sequence spaces. We need the following inequality in the sequel of the paper. For a, b 0 and 0 < p < 1, we have (a + b p a p + b p. (1 The double series m,n=1 x mn is called convergent if and only if the double sequence (s mn is convergent, where s mn = m,n i,j=1 x ij(m, n N (see [1]. A sequence x = (x mn is said to be double analytic if sup mn x mn 1/m+n <. The vector space of all double analytic sequences will be denoted by Λ. A sequence x = (x mn is called double gai sequence if ((m + n! x mn 1/m+n 0 as m, n. The double gai sequences will be denoted by χ. Let φ = {all finite sequences}. Consider a double sequence x = (x ij. The (m, n th section x [m,n] of the sequence is defined by x [m,n] = m,n i,j=0 x iji ij for all m, n N, where I ij denotes the double sequence whose only non zero term is a 1 (i+j! in the (i, j th place for each i, j N.

109 Vol. 8 The v invariant χ sequence spaces 103 An F K-space (or a metric space X is said to have AK property if (I mn is a Schauder basis for X. Or equivalently x [m,n] x. An F DK-space is a double sequence space endowed with a complete metrizable, locally convex topology under which the coordinate mappings x = (x k (x mn (m, n N are also continuous. If X is a sequence space, we give the following definitions: (i X = the continuous dual of X, (ii X α = { a = (a mn : m,n=1 a mn x mn <, for each x X }, (iii X β = { a = (a mn : m,n=1 a mn x mn is convergent, for each x X }, { (iv X γ = a = (a mn : sup mn 1 } M,N m,n=1 a mnx mn <, for each x X, { f(i mn : f X }, (v LetXbe an F K-space φ, then X f = (vi X δ = { a = (a mn : sup mn a mn x mn 1/m+n <, for each x X X α, X β, X γ are called α- (or Köthe-Toeplitz dual of X, β- (or generalized-köthe- Toeplitz dual of X, γ-dual of X, δ-dual of X respectively. X α is defined by Gupta and Kamptan [0]. It is clear that x α X β and X α X γ, but X β X γ does not hold, since the sequence of partial sums of a double convergent series need not to be bounded. The notion of difference sequence spaces (for single sequences was introduced by Kizmaz [30] as follows Z ( = {x = (x k w : ( x k Z}, for Z = c, c 0 and l, where x k = x k x k+1 for all k N. Here c, c 0 and l denote the classes of convergent,null and bounded sclar valued single sequences respectively. The difference space bv p of the classical space l p is introduced and studied in the case 1 p by BaŞar and Altay in [4] and in the case 0 < p < 1 by Altay and BaŞar in [43]. The spaces c (, c 0 (, l ( and bv p are Banach spaces normed by x = x 1 + sup k 1 x k and x bvp }. = ( k=1 x k p 1/p, (1 p <. Later on the notion was further investigated by many others. following difference double sequence spaces defined by Z ( = { x = (x mn w : ( x mn Z }, We now introduce the where Z = Λ, χ and x mn = (x mn x mn+1 (x m+1n x m+1n+1 = x mn x mn+1 x m+1n + x m+1n+1 for all m, n N.. Definitions and preliminaries Definition.1. A sequence X is v-invariant if X v = X where X v = {x = (x mn : (v mn x mn X}, where X = Λ and χ. Definition.. We say that a sequence space (X is v-invariant if v (X = (X. ( Definition.3. Let A = a mn k,l denotes a four dimensional summability method that maps the complex double sequences x into the double sequence Ax where the k, l-th term to Ax is as follows:

110 104 Nagarajan Subramanian and Umakanta Misra No. 4 (Ax kl = m=1 n=1 amn kl x mn, such transformation is said to be nonnegative if a mn kl is nonnegative. The notion of regularity for two dimensional matrix transformations was presented by Silverman [40] and Toeplitz [41]. Following Silverman and Toeplitz, Robison and Hamilton presented the following four dimensional analog of regularity for double sequences in which they both added an adiditional assumption of boundedness. This assumption was made because a double sequence which is P -convergent is not necessarily bounded. 3. Main results Lemma 3.1. sup mn x mn x mn+1 x m+1n+1 + x m+1n+1 1/m+n < if and only if (i sup (mn 1 x mn 1/m+n <, mn (ii sup x mn (mn (mn x mn+1 mn (mn (m + 1n 1 x m+1n + (mn (m + 1n x m+1n+1 1/m+n <. If we consider Lemma (3.1, then we have the following result. Corollary 3.1. sup (mn v mn (mn + 1 v mn+1 mn w mn w mn+1 (m + 1n v m+1n + (m + 1n + 1 v m+1n+1 1/m+n <, w m+1n w m+1n+1 if and only if (i 1/m+n sup v mn mn w mn <, (ii sup (mn v mn v mn+1 v m+1n + v 1/m+n m+1n+1 mn w mn w mn+1 w m+1n w m+1n+1 <. ( Theorem 3.1. ( w Λ v Λ if and only if the matrix A = (a mn kl maps Λ into Λ where v i+1j+1 + v i+1j + v ij+1 v ij, v i+1j+1 w i+1j+1, if m, n=i+1, j+1, a mn v kl = i+1j w i+1j, if m, n=i+1, j, ( v ij+1 w ij+1, if m, n i, j+1, 0, if m, n i+, j+. Proof. Let y Λ. Define x ij = i m=1 j n=1 y mn 1/m+n w pq.

111 Vol. 8 The v invariant χ sequence spaces 105 Then w ij x ij w ij+1 x ij+1 w i+1j x i+1j + w i+1j+1 x i+1j+1 Λ. Hence wx Λ, so by assumption v x Λ. It follows that Ay Λ. This shows that A = (a mn kl maps Λ into Λ. ( Let x w Λ, hence (w ij x ij w ij+1 x ij+1 w i+1j x i+1j + w i+1j+1 x i+1j+1 Λ. Then also y = (w i 1j 1 x i 1j 1 w i 1j x i 1j w ij 1 x ij 1 + w ij x ij Λ, where w 1 = x 1 = 0. By ( assumption we have Ay Λ. Hence x v Λ. This completes the proof. Theorem 3.. Let X and Y be sequence spaces and assume that X is such that a sequence x 11 x 1 x 13 x 1n 0 x 1 x x 3 x n 0. belongs to X if and only if the sequence x m1 x m x m3 x mn x 1 x x 3 x n 0 x 31 x 3 x 33 x 3n 0.. x m1 x m x m3 x mn Then we have w (X v (Y if and only if the matrix A = (a mn kl maps X into Y, where A = (a mn kl is defined by equation (. The proof is very similar to that of Theorem (3.3. ( Corollary 3.. We have ( w Λ v Λ if and only if sup mn mn v mn mn + 1 v mn+1 m + 1n v m+1n + m + 1n + 1 v m+1n+1 w mn w mn+1 w m+1n w m+1n+1 1/m+n <. Proof. This follows from Theorem 3.3, the well known characterization of matrices mapping Λ into Λ and corollary 3.. ( Corollary 3.3. We have ( w χ v χ if and only if ((m + n! mn v mn mn + 1 v mn+1 m + 1n v m+1n + m + 1n + 1 v m+1n+1 w mn w mn+1 w m+1n w m+1n+1 1/m+n 0 as m, n. Proof. This follows from Theorem 3.5, the well-known characterization of matrices mapping χ into χ and corollary 3.. If we consider corollary 3.5 and corollary 3.6, then we have necessary and sufficient conditions for the v-invariant of ( Λ and ( χ. References [1] Apostol, T. Apostol, Mathematical Analysis, Addison-wesley, London, [] Basarir, M. Basarir and O. Solancan, On some double sequence spaces, J. Indian Acad. Math., 1(1999, No.,

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114 Scientia Magna Vol. 8 (01, No. 4, Weakly generalized compactness in intuitionistic fuzzy topological spaces P. Rajarajeswari and R. Krishna Moorthy Department of Mathematics, Chikkanna Government Arts College, Tirupur, 64160, Tamil Nadu, India Department of Mathematics, Kumaraguru College of Technology, Coimbatore, , Tamil Nadu, India p.rajarajeswari9@gmail.com krishnamoorthykct@gmail.com Abstract The aim of this paper is to introduce and study the concepts of weakly generalized compact space, almost weakly generalized compact space using intuitionistic fuzzy weakly generalized open sets and nearly weakly generalized compact space using intuitionistic fuzzy weakly generalized closed sets in intuitionistic fuzzy topological space. Some of their properties are explored. Keywords Intuitionistic fuzzy topology, intuitionistic fuzzy weakly generalized compact spa -ce, intuitionistic fuzzy almost weakly generalized compact space and intuitionistic fuzzy nea -rly weakly generalized compact space. 000 Mathematics Subject Classification: 54A40, 03E7 1. Introduction Fuzzy set (F S as proposed by Zadeh [13] in 1965, is a framework to encounter uncertainty, vagueness and partial truth and it represents a degree of membership for each member of the universe of discourse to a subset of it. After the introduction of fuzzy topology by Chang [3] in 1968, there have been several generalizations of notions of fuzzy sets and fuzzy topology. By adding the degree of non-membership to F S, Atanassov [1] proposed intuitionistic fuzzy set (IF S in 1986 which looks more accurate to uncertainty quantification and provides the opportunity to precisely model the problem based on the existing knowledge and observations. In 1997, Coker [4] introduced the concept of intuitionistic fuzzy topological space. In this paper, we introduce a new class of intuitionistic fuzzy topological space called weakly generalized compact space, almost weakly generalized compact space using intuitionistic fuzzy weakly generalized open sets and nearly weakly generalized compact space using intuitionistic fuzzy weakly generalized closed sets and study some of their properties.

115 Vol. 8 Weakly generalized compactness in intuitionistic fuzzy topological spaces 109. Preliminaries Definition.1. [1] Let X be a non empty fixed set. An intuitionistic fuzzy set (IF S in short A in X is an object having the form A = { x, µ A (x, ν A (x : x X} where the functions µ A (x : X [0, 1] and ν A (x : X [0, 1] denote the degree of membership (namely µ A (x and the degree of non-membership (namely ν A (x of each element x X to the set A, respectively, and 0 µ A (x + ν A (x 1 for each x X. Definition.. [1] Let A and B be IF Ss of the forms A = { x, µ A (x, ν A (x : x X} and B = { x, µ B (x, ν B (x : x X}. Then (i A B if and only if µ A (x µ B (x and ν A (x ν B (x for all x X. (ii A = B if and only if A B and B A. (iii A c = { x, ν A (x, µ A (x : x X}. (iv A B = { x, µ A (x µ B (x, ν A (x ν B (x : x X}. (v A B = { x, µ A (x µ B (x, ν A (x ν B (x : x X}. For the sake of simplicity, the notation A = x, µ A, ν A shall be used instead of A = { x, µ A (x, ν A (x : x X}. Also for the sake of simplicity, we shall use the notation A = x, (µ A, µ B, (ν A, ν B instead of A = x, (A/µ A, B/µ B, (A/ν A, B/ν B. The intuitionistic fuzzy sets 0 = { x, 0, 1 : x X} and 1 = { x, 1, 0 : x X} are the empty set and the whole set of X, respectively. Definition.3. [4] An intuitionistic fuzzy topology (IF T in short on a non empty set X is a family τ of IF Ss in X satisfying the following axioms: (i 0, 1 τ. (ii G 1 G τ for any G 1, G τ. (iii G i τ for any arbitrary family {G i : i J} τ. In this case, the pair (X, τ is called an intuitionistic fuzzy topological space (IF T S in short and any IF S in τ is known as an intuitionistic fuzzy open set (IF OS in short in X. The complement A c of an IF OSA in an IF T S (X, τ is called an intuitionistic fuzzy closed set (IF CS in short in X. Definition.4. [4] Let (X, τ be an IF T S and A = x, µ A, ν A be an IF S in X. Then the intuitionistic fuzzy interior and an intuitionistic fuzzy closure are defined by int(a = {G/G is an IFOS in X and G A}, cl(a = {K/K is an IFCS in X and A K}. Note that for any IFS A in (X, τ, we have cl(a c = (int(a c and int(a c = (cl(a c. Definition.5. [5] An IF S A = { x, µ A (x, ν A (x : x X} in an IF T S (X, τ is said to be an intuitionistic fuzzy weakly generalized closed set (IF W GCS in short if cl(int(a U whenever A U and U is an IF OS in X. The family of all IF W GCSs of an IF T S (X, τ is denoted by IF W GC(X. Definition.6. [5] An IF S A = { x, µ A (x, ν A (x : x X} is said to be an intuitionistic fuzzy weakly generalized open set (IF W GOS in short in (X, τ if the complement A c is an IF W GCS in (X, τ. The family of all IF W GOSs of an IF T S (X, τ is denoted by IF W GO(X.

116 110 P. Rajarajeswari and R. Krishna Moorthy No. 4 Result.1. [5] Every IF CS, IF αcs, IF GCS, IF RCS, IF P CS, IF αgcs is an IF W GCS but the converses need not be true in general. Definition.7. [6] Let (X, τ be an IFTS and A = x, µ A, ν A be an IF S in X. Then the intuitionistic fuzzy weakly generalized interior and an intuitionistic fuzzy weakly generalized closure are defined by wgint(a = {G/G is an IFWGOS in X and G A}, wgcl(a = {K/K is an IFWGCS in X and A K}. Definition.8. Let f : (X, τ (Y, σ be a mapping from an IF T S (X, τ into an IF T S (Y, σ. Then f is said to be (i Intuitionistic fuzzy weakly generalized continuous [7] (IF W G continuous in short if f 1 (B is an IF W GOS in X for every IF OS B in Y. (ii Intuitionistic fuzzy quasi weakly generalized continuous [8] (IF quasi W G continuous in short if f 1 (B is an IF OS in X for every IF W GOS B in Y. (iii Intuitionistic fuzzy weakly generalized iresolute [6] (IF W G irresolute in short if f 1 (B is an IF W GOS in X for every IF W GOS B in Y. (iv Intuitionistic fuzzy perfectly weakly genralized continuous continuous [10] (IF perfectly W G continuous in short if f 1 (B is an intuitionistic fuzzy clopen set in X for every IF W GCS B in Y. (v Intuitionistic fuzzy weakly generalized * open mapping [9] (IF W G OM in short if f(b is an IF W GOS in Y for every IF W GOS B in X. (vi Intuitionistic fuzzy contra weakly generalized continuous [11] (IF contra W G continuous in short if f 1 (B is an IF W GCS in X for every IF OS B in Y. (vii Intuitionistic fuzzy contra weakly generalized irresolute [1] (IF contra W G irresolute in short if f 1 (B is an IF W GCS in X for every IF W GOS B in Y. Definition.9. [] Let (X, τ be an IF T S. A family { x, µ Gi, ν Gi : i I} of IF OSs in X satisfying the condition 1 = { x, µ Gi, ν Gi : i I} is called an intuitionistic fuzzy open cover of X. Definition.10. [] A finite sub family of an intuitionistic fuzzy open cover { x, µ Gi, ν Gi : i I} of X which is also an intuitionistic fuzzy open cover of X is called a finite sub cover of { x, µ Gi, ν Gi : i I}. Definition.11. [] An IF T S (X, τ is called intuitionistic fuzzy compact if every intuitionistic fuzzy open cover of X has a finite sub cover. Definition.1. [] An IF T S (X, τ is called intuitionistic fuzzy Lindelof if each intuitionistic fuzzy open cover of X has a countable sub cover for X. Definition.13. [] An IF T S (X, τ is called intuitionistic fuzzy countable compact if each countable intuitionistic fuzzy open cover of X has a finite sub cover for X. Definition.14. An IF T S (X, τ is said to be (i Intuitionistic fuzzy S-closed [] if each intuitionistic fuzzy regular closed cover of X has a finite sub cover for X. (ii Intuitionistic fuzzy S-Lindelof [] if each intuitionistic fuzzy regular closed cover of X has a countable sub cover for X.

117 Vol. 8 Weakly generalized compactness in intuitionistic fuzzy topological spaces 111 (iii Intuitionistic fuzzy countable S-closed [] if each countable intuitionistic fuzzy regular closed cover of X has a finite sub cover for X. Definition.15. An IF T S (X, τ is said to be (i Intuitionistic fuzzy strongly S-closed [] if each intuitionistic fuzzy closed cover of X has a finite sub cover for X. (ii Intuitionistic fuzzy strongly S-Lindelof [] if each intuitionistic fuzzy closed cover of X has a countable sub cover for X. (iii Intuitionistic fuzzy countable strongly S-closed [] if each countable intuitionistic fuzzy closed cover of X has a finite sub cover for X. Definition.16. An IF T S (X, τ is sad to be (i Intuitionistic fuzzy almost compact [] if each intuitionistic fuzzy open cover of X has a finite sub cover the closure of whose members cover X. (ii Intuitionistic fuzzy almost Lindelof [] if each intuitionistic fuzzy open cover of X has a countable sub cover the closure of whose members cover X. (iii Intuitionistic fuzzy countable almost compact [] if each countable intuitionistic fuzzy open cover of X has a finite sub cover the closure of whose members cover X. 3. Intuitionistic fuzzy weakly generalized compact spaces In this section, we introduce a new class of intuitionistic fuzzy topological spaces called weakly generalized compact spaces, almost weakly generalized compact spaces using intuitionistic fuzzy weakly generalized open sets and nearly weakly generalized compact spaces using intuitionistic fuzzy weakly generalized closed sets and study some of their properties. Definition 3.1. Let (X, τ be an IF T S. A family { x, µ Gi, ν Gi : i I} of IF W GOSs in X satisfying the condition 1 = { x, µ Gi, ν Gi : i I} is called an intuitionistic fuzzy weakly generalized open cover of X. Definition 3.. A finite sub family of an intuitionistic fuzzy weakly generalized open cover { x, µ Gi, ν Gi : i I} of X which is also an intuitionistic fuzzy weakly generalized open cover of X is called a finite sub cover of { x, µ Gi, ν Gi : i I}. Definition 3.3. An intuitionistic fuzzy set A of an IF T S(X, τ is said to be intuitionistic fuzzy weakly generalized compact relative to X if every collection {A i : i I} of intuitionistic fuzzy weakly generalized open subset of X such that A {A i : i I}, there exists a finite subset I 0 of I such that A {A i : i I 0 }. Definition 3.4. Let (X, τ be an IF T S. A family { x, µ Gi, ν Gi : i I} of IFWGCSs in X satisfying the condition 1 = { x, µ Gi, ν Gi : i I} is called an intuitionistic fuzzy weakly generalized closed cover of X. Definition 3.5. An IF T S (X, τ is said to be intuitionistic fuzzy weakly generalized compact if every intuitionistic fuzzy weakly generalized open cover of X has a finite sub cover. Definition 3.6. An IF T S (X, τ is said to be intuitionistic fuzzy weakly generalized Lindelof if each intuitionistic fuzzy weakly generalized open cover of X has a countable sub cover for X.

118 11 P. Rajarajeswari and R. Krishna Moorthy No. 4 Definition 3.7. An IF T S (X, τ is said to be intuitionistic fuzzy countable weakly generalized compact if each countable intuitionistic fuzzy weakly generalized open cover of X has a finite sub cover for X. Definition 3.8. An IF T S (X, τ is said to be intuitionistic fuzzy nearly weakly generalized compact if each intuitionistic fuzzy weakly generalized closed cover of X has a finite sub cover. Definition 3.9. An IF T S (X, τ is said to be intuitionistic fuzzy nearly weakly generalized Lindelof if each intuitionistic fuzzy weakly generalized closed cover of X has a countable sub cover for X. Definition An IF T S (X, τ is said to be intuitionistic fuzzy nearly countable weakly generalized compact if each countable intuitionistic fuzzy weakly generalized closed cover of X has a finite sub cover for X. Definition An IF T S (X, τ is said to be intuitionistic fuzzy almost weakly generalized compact if each intuitionistic fuzzy weakly generalized open cover of X has a finite sub cover the closure of whose members cover X. Definition 3.1. An IF T S (X, τ is said to be intuitionistic fuzzy almost weakly generalized Lindelof if each intuitionistic fuzzy weakly generalized open cover of X has a countable sub cover the closure of whose members cover X. Definition An IF T S (X, τ is said to be intuitionistic fuzzy almost countable weakly generalized compact if each countable intuitionistic fuzzy weakly generalized open cover of X has a finite sub cover the closure of whose members cover X. Definition A crisp subset B of an IF T S (X, τ is said to be intuitionistic fuzzy weakly generalized compact if B is intuitionistic fuzzy weakly generalized compact as intuitionistic fuzzy subspace of X. Theorem 3.1. An intuitionistic fuzzy weakly generalized closed crisp subset of an intuitionistic fuzzy weakly generalized compact space is intuitionistic fuzzy weakly generalized compact relative to X. Proof. Let A be an intuitionistic fuzzy weakly generalized closed crisp subset of an intuitionistic fuzzy weakly generalized compact space (X, τ. Then A c is an IF W GOS in X. Let S be a cover of A by IF W GOS in X. Then, the family {S, A c } is an intuitionistic fuzzy weakly generalized open cover of X. Since X is an intuitionistic fuzzy weakly generalized compact space, it has finite sub cover say {G 1, G... G n }. If this sub cover contains A c, we discard it. Otherwise leave the sub cover as it is. Thus we obtained a finite intuitionistic fuzzy weakly generalized open sub cover of A. Therefore A is intuitionistic fuzzy weakly generalized compact relative to X. Theorem 3.. Let f : (X, τ (Y, σ be an IF W G continuous mapping from an IF T S (X, τ onto an IF T S (Y, σ. If (X, τ is intuitionistic fuzzy weakly generalized compact, then (Y, σ is intuitionistic fuzzy compact. Proof. Let {A i : i I} be an intuitionistic fuzzy open cover of (Y, σ. Then 1 = i I A i. From the relation, 1 = f 1 ( i I A i follows that 1 = i I f 1 (A i, so { f 1 (A i : i I } is an intuitionistic fuzzy weakly generalized open cover of (X, τ. Since (X, τ is intuitionistic fuzzy weakly generalized compact, there exists a finite sub cover say, {f 1 (A 1, f 1 (A,... f 1 (A n }.

119 Vol. 8 Weakly generalized compactness in intuitionistic fuzzy topological spaces 113 Therefore 1 = n i=1 f 1 (A i Hence ( n 1 = f f 1 (A i = i=1 n f ( f 1 (A i = i=1 n A i. That is, {A 1, A,... A n } is a finite sub cover of (Y, σ. Hence (Y, σ is intuitionistic fuzzy compact. Corollary 3.1. Let f : (X, τ (Y, σ be an IF W G continuous mapping from an IF T S (X, τ onto an IF T S (Y, σ. If (X, τ is intuitionistic fuzzy weakly generalized Lindelof, then (Y, σ is intuitionistic fuzzy Lindelof. Proof. Obvious. Theorem 3.3. Let f : (X, τ (Y, σ be an IF quasi W G continuous mapping from an IF T S (X, τ onto an IF T S (Y, σ. If (X, τ is intuitionistic fuzzy compact, then (Y, σ is intuitionistic fuzzy weakly generalized compact. Proof. Let {A i : i I} be an intuitionistic fuzzy weakly generalized open cover of (Y, σ. Then 1 = i I A i. From the relation, 1 = f 1 ( i I A i follows that 1 = i I f 1 (A i, so { f 1 (A i : i I } is an intuitionistic fuzzy open cover of (X, τ. Since (X, τ is intuitionistic fuzzy compact, there exists a finite sub cover say, { f 1 (A 1, f 1 (A,... f 1 (A n }. Therefore 1 = n i=1 f 1 (A i. Hence ( n n 1 = f f 1 (A i = f ( f 1 (A i n = A i. i=1 i=1 That is, {A 1, A,... A n } is a finite sub cover of (Y, σ. Hence (Y, σ is intuitionistic fuzzy weakly generalized compact. Corollary 3.. Let f : (X, τ (Y, σ be an IF perfectly W G continuous mapping from an IF T S (X, τ onto an IF T S (Y, σ. If (X, τ is intuitionistic fuzzy compact, then (Y, σ is intuitionistic fuzzy weakly generalized compact. Proof. Obvious. Theorem 3.4. If f : (X, τ (Y, σ is an IF W G irrresolute mapping and an intuitionistic fuzzy subset B of an IF T S (X, τ is intuitionistic fuzzy weakly generalized compact relative to an IF T S (X, τ, then the image f(b is intuitionistic fuzzy weakly generalized compact relative to (Y, σ. Proof. Let {A i : i I} be any collection IF W GOSs of (Y, σ such that f(b {A i : i I}. Since f is IF W G irresolute, B { f 1 (A i : i I } where f 1 (A i is intuitionistic fuzzy weakly generalized open cover in (X, τ for each i. Since B is intuitionistic fuzzy weakly generalized compact relative to (X, τ, there exists a finite subset I 0 of I such that B { f 1 (A i : i I 0 }. Therefore f(b {Ai : I I 0 }. Hence f(b is intuitionistic fuzzy weakly generalized compact relative to (Y, σ. Theorem 3.5. Let f : (X, τ (Y, σ be an IF W G irresolute mapping from an IF T S (X, τ onto an IF T S (Y, σ. If (X, τ is intuitionistic fuzzy nearly weakly generalized compact, then (Y, σ is intuitionistic fuzzy nearly weakly generalized compact. Proof. Let {A i : i I} be an intuitionistic fuzzy weakly generalized closed cover of (Y, σ. Then 1 = i I A i. From the relation, 1 = f 1 ( i I A i follows that 1 = i I f 1 (A i, i=1 i=1

120 114 P. Rajarajeswari and R. Krishna Moorthy No. 4 so { f 1 (A i : i I } is an intuitionistic fuzzy weakly generalized closed cover of (X, τ. Since (X, τ is intuitionistic fuzzy nearly weakly generalized compact, there exists a finite sub cover say, {f 1 (A 1, f 1 (A,... f 1 (A n }. Therefore 1 = n i=1 f 1 (A i. Hence ( n n 1 = f f 1 (A i = f ( f 1 (A i n = A i. i=1 That is, {A 1, A,... A n } is a finite sub cover of (Y, σ. Hence (Y, σ is intuitionistic fuzzy nearly weakly generalized compact. Corollary 3.3. Let f : (X, τ (Y, σ be an IF W G irresolute mapping from an IF T S (X, τ onto an IF T S (Y, σ. Then the following statements hold. (i If (X, τ is intuitionistic fuzzy weakly generalized compact, then (Y, σ is intuitionistic fuzzy weakly generalized compact. (ii If (X, τ is intuitionistic fuzzy countable weakly generalized compact, then (Y, σ is intuitionistic fuzzy countable weakly generalized compact. (iii If (X, τ is intuitionistic fuzzy weakly generalized Lindelof, then (Y, σ is intuitionistic fuzzy weakly generalized Lindelof. (iv If (X, τ is intuitionistic fuzzy nearly countable weakly generalized compact, then (Y, σ is intuitionistic fuzzy nearly countable weakly generalized compact. (v If (X, τ is intuitionistic fuzzy nearly weakly generalized Lindelof, then (Y, σ is intuitionistic fuzzy nearly weakly generalized Lindelof. Proof. Obvious. Theorem 3.6. Let f : (X, τ (Y, σ be an IF contra W G continuous mapping from an IF T S (X, τ onto an IF T S (Y, σ. If (X, τ is intuitionistic fuzzy nearly weakly generalized compact, then (Y, σ is intuitionistic fuzzy compact. Proof. Let {A i : i I} be an intuitionistic fuzzy open cover of (Y, σ. Then 1 = i I A i. From the relation, 1 = f 1 ( i I A i follows that 1 = i I f 1 (A i, so { f 1 (A i : i I } is an intuitionistic fuzzy weakly generalized closed cover of (X, τ. Since (X, τ is intuitionistic fuzzy nearly weakly generalized compact, there exists a finite sub cover say, {f 1 (A 1, f 1 (A,.. f 1 (A n }. Therefore 1 = n i=1 f 1 (A i. Hence ( n n 1 = f f 1 (A i = f ( f 1 (A i n = A i. i=1 That is, {A 1, A,... A n } is a finite sub cover of (Y, σ. Hence (Y, σ is intuitionistic fuzzy compact. Corollary 3.4. Let f : (X, τ (Y, σ be an IF contra W G continuous mapping from an IF T S (X, τ onto an IF T S (Y, σ. If (X, τ is intuitionistic fuzzy nearly weakly generalized Lindelof, then (Y, σ is intuitionistic fuzzy Lindelof. Proof. Obvious. Theorem 3.7. Let f : (X, τ (Y, σ be an IF contra W G continuous mapping from an IF T S (X, τ onto an IF T S (Y, σ. If (X, τ is intuitionistic fuzzy weakly generalized compact, then (Y, σ is intuitionistic fuzzy strongly S-closed. Proof. Let {A i : i I} be an intuitionistic fuzzy closed cover of (Y, σ. Then 1 = i I A i. From the relation, 1 = f 1 ( i I A i follows that 1 = i I f 1 (A i, so { f 1 (A i : i I } is i=1 i=1 i=1 i=1

121 Vol. 8 Weakly generalized compactness in intuitionistic fuzzy topological spaces 115 an intuitionistic fuzzy weakly generalized open cover of (X, τ. Since(X, τ is intuitionistic fuzzy weakly generalized compact, there exists a finite sub cover say, {f 1 (A 1, f 1 (A,... f 1 (A n }. Therefore 1 = n i=1 f 1 (A i. Hence n n n 1 = f( f 1 (A i = f(f 1 (A i = A i. i=1 i=1 That is, {A 1, A,... A n } is a finite sub cover of (Y, σ. Hence (Y, σ is intuitionistic fuzzy strongly S-closed. Corollary 3.5. Let f : (X, τ (Y, σ be an IF contra W G continuous mapping from an IF T S (X, τ onto an IF T S (Y, σ. Then the following statements hold. (i If (X, τ is intuitionistic fuzzy weakly generalized Lindelof, then (Y, σ is intuitionistic fuzzy strongly S-Lindelof. (ii If (X, τ is intuitionistic fuzzy countable weakly generalized compact, then (Y, σ is intuitionistic fuzzy countable strongly S-Closed. Proof. Obvious. Theorem 3.8. Let f : (X, τ (Y, σ be an IF contra W G continuous mapping from an IF T S (X, τ onto an IF T S (Y, σ. If (X, τ is intuitionistic fuzzy weakly generalized compact (respectively, intuitionistic fuzzy weakly generalized Lindelof, intuitionistic fuzzy countable weakly generalized compact, then (Y, σ is intuitionistic fuzzy S-closed (respectively, intuitionistic fuzzy S-Lindelof, intuitionistic fuzzy countable S-closed. Proof. It follows from the statement that each IF RCS is an IF CS. Theorem 3.9. Let f : (X, τ (Y, σ be an IF contra W G continuous mapping from an IF T S (X, τ onto an IF T S (Y, σ. If (X, τ is intuitionistic fuzzy nearly weakly generalized compact, then (Y, σ is intuitionistic fuzzy almost compact. Proof. Let {A i : i I} be an intuitionistic fuzzy open cover of (Y, σ. Then 1 = i I A i. It follows that 1 = i I cl(a i. From the relation, 1 = f 1 ( i I cl(a i follows that 1 = i I f 1 cl(a i, so {f 1 cl(a i : i I} is an intuitionistic fuzzy weakly generalized closed cover of (X, τ. Since (X, τ is intuitionistic fuzzy nearly weakly generalized compact, there exists a finite sub cover say, {f 1 cl(a 1, f 1 cl(a,... f 1 cl(a n }. Therefore 1 = n i=1 f 1 cl(a i. Hence n n n 1 = f( f 1 cl(a i = f(f 1 cl(a i = cl(a i. i=1 i=1 Hence (Y, σ is intuitionistic fuzzy almost compact. Corollary 3.6. Let f : (X, τ (Y, σ be an IF contra WG continuous mapping from an IF T S (X, τ onto an IF T S (Y, σ. Then the following statements hold. (i If (X, τ is intuitionistic fuzzy nearly weakly generalized Lindelof, then (Y, σ is intuitionistic fuzzy almost Lindelof. (ii If (X, τ is intuitionistic fuzzy nearly countable weakly generalized compact, then (Y, σ is intuitionistic fuzzy countable almost compact. Proof. Obvious. Theorem Let f : (X, τ (Y, σ be an IF W G open bijective mapping from an IF T S (X, τ onto an IF T S (Y, σ. If (Y, σ is intuitionistic fuzzy weakly generalized compact, then (X, τ is intuitionistic fuzzy weakly generalized compact. i=1 i=1

122 116 P. Rajarajeswari and R. Krishna Moorthy No. 4 Proof. Let {A i : i I} be an intuitionistic fuzzy weakly generalized open cover of (X, τ. Then 1 = i I A i. From the relation, 1 = f ( i I A i follows that 1 = i I f (A i, so {f (A i : i I} is an intuitionistic fuzzy weakly generalized open cover of (Y, σ. Since (Y, σ is intuitionistic fuzzy weakly generalized compact, there exists a finite sub cover say, {f(a 1, f(a,...f(a n }. Therefore 1 = n i=1 f (A i. Hence ( n n n 1 = f 1 f (A i = f 1 (f (A i = A i. i=1 That is, {A 1, A,... A n } is a finite sub cover of (X, τ. Hence (X, τ is intuitionistic fuzzy compact. Theorem Let f : (X, τ (Y, σ be an IF contra W G irresolute mapping from an IF T S (X, τ onto an IF T S (Y, σ. If (X, τ is intuitionistic fuzzy nearly weakly generalized compact, then (Y, σ is intuitionistic fuzzy weakly generalized compact. Proof. Let {A i : i I} be an intuitionistic fuzzy weakly generalized open cover of (Y, σ. Then 1 = i I A i. From the relation, 1 = f 1 ( i I A i follows that 1 = i I f 1 (A i, so {f 1 (A i : i I} is an intuitionistic fuzzy weakly generalized closed cover of (X, τ. Since (X, τ is intuitionistic fuzzy nearly weakly generalized compact, there exists a finite sub cover say, {f 1 (A 1, f 1 (A,... f 1 (A n }. Therefore 1 = n i=1 f 1 (A i. Hence n n n 1 = f( f 1 (A i = f(f 1 (A i = A i. i=1 i=1 i=1 That is, {A 1, A,... A n } is a finite sub cover of (Y, σ. Hence (Y, σ is intuitionistic fuzzy weakly generalized compact. Corollary 3.7. Let f : (X, τ (Y, σ be an IF contra W G irresolute mapping from an IF T S (X, τ onto an IF T S (Y, σ. Then the following statements hold. (i If (X, τ is intuitionistic fuzzy nearly weakly generalized Lindelof, then (Y, σ is intuitionistic fuzzy weakly generalized Lindelof. (ii If (X, τ is intuitionistic fuzzy nearly countable weakly generalized compact, then (Y, σ is intuitionistic fuzzy countable weakly generalized compact. (iii If (X, τ is intuitionistic fuzzy weakly generalized compact, then (Y, σ is intuitionistic fuzzy nearly weakly generalized compact. (iv If (X, τ is intuitionistic fuzzy weakly generalized Lindelof, then (Y, σ is intuitionistic fuzzy nearly weakly generalized Lindelof. (v If (X, τ is intuitionistic fuzzy countable weakly generalized compact, then (Y, σ is intuitionistic fuzzy nearly countable weakly generalized compact. Proof. Obvious. Theorem 3.1. Let f : (X, τ (Y, σ be an IF contra W G irresolute mapping from an IF T S (X, τ onto an IF T S (Y, σ. If (X, τ is intuitionistic fuzzy nearly weakly generalized compact, then (Y, σ is intuitionistic fuzzy almost weakly generalized compact. Proof. Let {A i : i I} be an intuitionistic fuzzy weakly generalized open cover of (Y, σ. Then 1 = i I A i. It follows that 1 = i I cl(a i. From the relation, 1 = f 1 ( i I cl(a i follows that 1 = i I f 1 cl (A i, so { f 1 cl (A i : i I } is an intuitionistic fuzzy weakly generalized closed cover of (X, τ. Since (X, τ is intuitionistic fuzzy nearly weakly generalized i=1 i=1

123 Vol. 8 Weakly generalized compactness in intuitionistic fuzzy topological spaces 117 compact, there exists a finite sub cover say, { f 1 cl (A 1, f 1 cl (A,... f 1 cl (A n }. Therefore 1 = n i=1 f 1 cl (A i. Hence ( n n 1 = f f 1 cl (A i = f ( f 1 cl (A i n = cl(a i. i=1 i=1 Hence (Y, σ is intuitionistic fuzzy almost weakly generalized compact. Corollary 3.8. Let f : (X, τ (Y, σ be an IF contra W G irresolute mapping from an IF T S (X, τ onto an IF T S (Y, σ. Then the following statements hold. (i If (X, τ is intuitionistic fuzzy nearly weakly generalized Lindelof, then (Y, σ is intuitionistic fuzzy almost weakly generalized Lindelof. (ii If (X, τ is intuitionistic fuzzy nearly countable weakly generalized compact, then (Y, σ is intuitionistic fuzzy almost countable weakly generalized compact. Proof. Obvious. i=1 References [1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 0(1986, [] B. Krsteska and E. Ekici, Intuitionistic fuzzy contra strong pre-continuity, Faculty of Sciences and Mathematics, University of Nis, Serbia, (007, [3] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl, 4(1968, [4] D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems, 88(1997, [5] P. Rajarajeswari and R. Krishna Moorthy, On intuitionistic fuzzy weakly generalized closed set and its applications, International Journal of Computer Applications, 7(011, [6] P. Rajarajeswari and R. Krishna Moorthy, Intuitionistic fuzzy weakly generalized irresolute mappings, Ultrascientist of Physical Sciences, 4(01, [7] P. Rajarajeswari and R. Krishna Moorthy, Intuitionistic fuzzy weakly generalized continuous mappings, Far East Journal of Mathematical Sciences, 66(01, [8] P. Rajarajeswari and R. Krishna Moorthy, Intuitionistic fuzzy quasi weakly generalized continuous, Scientia Magna, 3(01, [9] P. Rajarajeswari and R. Krishna Moorthy, Intuitionistic fuzzy weakly generalized closed mappings, Journal of Advanced Studies in Topology, 4(01, 0-7. [10] P. Rajarajeswari and R. Krishna Moorthy, Intuitionistic fuzzy perfectly weakly generalized continuous mappings, Notes on Intuitionistic Fuzzy Sets, 18(01, [11] P. Rajarajeswari and R. Krishna Moorthy, Intuitionistic fuzzy contra weakly generalized continuous mappings, Annals of Fuzzy Mathematics and Informatics, 13(013, [1] P. Rajarajeswari and R. Krishna Moorthy, Intuitionistic fuzzy contra weakly generalized irresolute mappings (submitted. [13] L. A. Zadeh, Fuzzy Sets, Information and control, 8(1965,

124 Scientia Magna Vol. 8 (01, No. 4, On normal and fantastic filters of BL-algebras Wei Wang, Xu Yang and Liu Miao Hua College of Electrical Engineering, Southwest JiaoTong University, Chengdu, , China Department of applied mathematics, Xi an Shiyou University, Xi an, , China College of Science, Air Force Univercity of Engineering, Xi an, , China wwmath@xsyu.edu.cn Abstract We further study the fantastic filter and normal filter of BL-algebras. By studying the equivalent condition of fantastic filter, we reveal the relation between fantastic filter and normal filter of BL-algebras and we solve two open problems that Under what suitable condition a normal filter becomes a fantastic filter? and (Extension property for a normal filter Under what suitable condition extension property for normal filter holds?. Keywords Non-classical logics, BL-algebras, filter, fantastic filter, normal filter. 1. Introduction The origin of BL-algebras is in Mathematical Logic. Hájek introduced BL-algebras as algebraic structures for his Basic Logic in order to investigate many-valued logic by algebraic means [3,9]. They play the role of Lindenbaum algebras from classical propositional calculus. The main example of a BL-algebra is the interval [0,1] endowed with the structure induced by a continuous t-norm. M V -algebras, Gödel algebras and Product algebras are the most known classes of BL-algebras [4]. In studying of algebras, filters theory plays an important role. From logical point of view, various filters correspond to various sets of provable formulae [11]. In [4], [5] and [8], the notions of prime filter, Boolean filter, implicative filters, normal filters, fantastic filters and positive implicative filters in BL-algebras were proposed, and the properties of the filters were investigated. In [], several different filters of residuated lattices and triangle algebras were defined and their mutual dependencies and connections were examined. In [1], [6] and [1], filters of pseudo M V -algebras, pseudo BL-algebras, pseudo effect algebras and pseudo hoops were further studied. We found that filters are a useful tool to obtain results of BL-algebras. 0 Corresponding author is Wei Wang

125 Vol. 8 On normal and fantastic filters of BL-algebras 119 In Section, we present some basic definitions and results on filters and BL-algebras, then we introduce the some kinds of filters in BL-algebras. In [8], Saeid and Motamed proposed two open problems that Under what suitable condition a normal filter becomes a fantastic filter? and (Extension property for a normal filter Under what suitable condition extension property for normal filter holds? in BL-algebras. In Section 3, by studying the equivalent condition of fantastic filter, we reveal the relation between fantastic filter and normal filter of BL-algebras and we solve the open problems.. Main results of summation formula Here we recall some definitions and results which will be needed. Definition.1. [3] A BL-algebra is an algebra (A,,,,, 0, 1 of type (,,,, 0, 0 such that (A,,, 0, 1 is a bounded lattice, (A,, 1 is a commutative monoid and the following conditions hold for all x, y, z A, (A1 x y z iff x y z, (A x y = x (x y, (A3 (x y (y x = 1. Proposition.1. [8] In a BL-algebras A, the following properties hold for all x, y, z A, (1 y (x z = x (y z, ( 1 x = x, (3 x y iff x y = 1, (4 x y = ((x y y ((y x x, (5 x y y z x z, (6 x y z x z y, (7 x y (z x (z y, (8 x y (y z (x z, (9 x (x y y. Definition.. [3] A filter of a BL-algebra A is a nonempty subset F of A such that for all x, y A, (F1 if x, y F, then x y F, (F if x F and x y, then y F. It is easy to prove the following equivalent conditions of filter in a BL-algebra. Proposition.. [3] Let F be a nonempty subset of a BL-algebra A. Then F is a filter of A if and only if the following conditions hold (1 1 F, ( x, x y F implies y F. A filter F of a BL-algebra A is proper if F A. Definition.3. [8] Let A be a BL-algebra. A filter F of A is called normal if for any x, y, z A, z ((y x x F and z F imply (x y y F. Definition.4. [8] Let F be a nonempty subset of a BL-algebra A. Then F is called a fantastic filter of A if for all x, y, z A, the following conditions hold (1 1 F,

126 10 Wei Wang, Xu Yang and Liu Miao Hua No. 4 ( z (y x F, z F implies ((x y y x F. Definition.5. [8] Let F be a nonempty subset of a BL-algebra A. Then F is called an implicative filter of A if for all x, y, z A, the following conditions hold (1 1 F, ( x (y z F, x y F imply x z F. Definition.6. [8] Let F be a nonempty subset of a BL-algebra A. Then F is called a positive implicative filter of A if for all x, y, z A, the following conditions hold (1 1 F, ( x ((y z y F, x F imply y F. Definition.7. [10] Let F be a filter of A. F is called an ultra filter of A if it satisfies x F or x F for all x A. Definition.8. [10] Let F be a filter of A. F is called an obstinate filter of A if it satisfies x / F and y / F implies x y F for all x, y A. Definition.9. [5] A filter F of A is called Boolean if x x F for any x A. Definition.10. [8] A proper filter F of A is prime if for all x, y A, x y F implies x F or y F. Theorem.1. [10] Let F be a filter of A. Then the following conditions are equivalent (1 F is an obstinate filter of A, ( F is an ultra filter of A, (3 F is a Boolean and prime filter of A. 3. The relation among the filters of BL-algebras On the relation between fantastic filter and normal filter of a BL-algebra, by far we have Theorem 3.1. [8] Let F be a fantastic filter of A. Then F is a normal filter of A. It is easy to find that the converse of the theorem is not true. In [8], there are two open problems that Under what suitable condition a normal filter becomes a fantastic filter? and (Extension property for a normal filter Under what suitable condition extension property for normal filter holds? In this section, we investigate the relation between fantastic filter and normal filter in a BL-algebras. After giving the equivalent conditions of fantastic filter, we present the relation between the two filters in a BL-algebra and solve the open problems. Theorem 3.. [8] Let F be an implicative filter of a BL-algebra A. Then F is a normal filter if and only if (x y x F implies x F for any x, y A. Theorem 3.3. [8] Let F be a filter of a BL-algebra A. Then F is a normal filter if only if (y x x F implies (x y y F for all x, y A. Theorem 3.4. [8] Let F be a filter of a BL-algebra A, if (x y x F implies x F for any x, y A, then F is a normal filter. Theorem 3.5. [8] Let F be a filter of a BL-algebra A, if x x F implies x F for any x A, then F is a normal filter. Theorem 3.6. [8] Let F be a filter of a BL-algebra A, if (x x x F for any x A, then F is a normal filter.

127 Vol. 8 On normal and fantastic filters of BL-algebras 11 Theorem 3.7. [8] Let F be a filter of a BL-algebra A. Then F is a normal filter if only if x F implies x F for all x A. Theorem 3.8. [8] Every normal and implicative filter of a BL-algebra is a positive implicative filter. Corollary 3.1. Every implicative filter satisfying (y x x F implies (x y y F for all x, y A of a BL-algebra A is a positive implicative filter. Theorem 3.9. Let f be a fuzzy filter of a BL-algebra A. Then the followings are equivalent (1 F is a fantastic filter, ( y x F implies ((x y y x F for any x, y A, (3 x x F for all x A, (4 x z F, y z F implies ((x y y z F for all x, y, z A. Proof. (1 (. Let F be a fantastic filter of A, and y x F. Then 1 (y x = y x F and 1 F, hence ((x y y x F. ( (1. Let a filter F satisfy the condition and let z (y x F and z F. Then y x F, therefore also ((x y y x F. ( (3. Let y = 0 in (. (3 (4. Similar to the proof of Theorem 4.4 [7]. (4 (. Obvious when z = x. Corollary 3.. [8] Each fantastic filter of A is a normal filter. By Theorem 3.9, we can give an answer to the first open problem presented in [8] in the following theorem. Theorem Every normal and implicative filter of a BL-algebra is a fantastic filter. Proof. Suppose y x F. We have x ((x y y x, thus (((x y y x y x y. Further, ((((xy y y x y (((x y y x (x y (((x y y x = ((x y y ((x y x y x. Since y x F, hence also ((((x y y x y (((x y y x F. By Theorem 3., we get ((x y y x F, and hence F is a fantastic filter by Theorem Theorem [7] Let F be a filter of a BL-algebra A. Then the followings are equivalent (1 F is an implicative filter, ( y (y x implies y x F for any x, y A, (3 z (y x implies (z y (z x F for any x, y, z A, (4 z (y (y x F and z F implies y x F for any x, y, z A, (5 x x x F for all x A. Theorem 3.1. [7] Let F be a filter of a BL-algebra A. Then the followings are equivalent (1 F is a positive implicative filter, ( (x y x F implies x F for any x, y A, (3 (x x x F for any x A. Theorem Every positive implicative filter of A is a fantastic filter. Proof. Suppose F is a positive implicative filter and for any x, y A, y x F. We have x ((x y y x, thus (((x y y x y x y. Further, ((((x y y x y (((x y y x (x y (((x y y x = ((x y y ((x y x y x. Since y x F, hence also ((((x y y

128 1 Wei Wang, Xu Yang and Liu Miao Hua No. 4 x y (((x y y x F. Thus we get ((x y y x F, and hence F is a fantastic filter. Corollary 3.3. Let F be an implicative filter of a BL-algebra A. Then F is a normal filter if only if F is a positive implicative filter of A. Corollary 3.4. Let F be a positive implicative filter of A. Then F is a normal filter. Theorem Every normal and obstinate filter of a BL-algebra is a fantastic filter. Proof. Suppose F is a normal and obstinate filter of a BL-algebra A, then for any x A, if x F, then x F since F is a normal filter, by x x x, thus x x F. Further, if x / F, then x / F, since F is an obstinate filter, then x x F, and hence F is a fantastic filter. By Theorem.1, we have Theorem Every normal and ultra filter of a BL-algebra is a fantastic filter. Theorem Every normal, Boolean and prime filter of a BL-algebra is a fantastic filter. Combining all the above results, we can give answers to the first open problem presented in [8] respectively. Theorem (1 Every normal filter of a MV -algebra is equivalent to a fantastic filter. ( Every normal and implicative filter of a BL-algebra is a fantastic filter. (3 Every normal and obstinate filter of a BL-algebra is a fantastic filter. (4 Every normal and ultra filter of a BL-algebra is a fantastic filter, (5 Every normal, Boolean and prime filter of a BL-algebra is a fantastic filter. For the second open problem presented in [8], since the extension property for a fantastic filter holds [7], and further by the above results, we have the following theorem. Theorem (Extension property for a normal filter Under one of the following conditions extension property for normal filter holds (1 If A is a MV -algebra. ( F is a normal and implicative filter of A. (3 F is a normal and obstinate filter of A. (4 F is a normal and ultra filter of A, (5 F is a normal, Boolean and prime filter of A. 4. Conclusion By studying the equivalent condition of fantastic filter, we reveal the relation between fantastic filter and normal filter of BL-algebras and we solve an open problem that Under what suitable condition a normal filter becomes a fantastic filter? and (Extension property for a normal filter Under what suitable condition extension property for normal filter holds?. In the future, we will extend the corresponding filter theory and study the congruence relations induced by the filters.

129 Vol. 8 On normal and fantastic filters of BL-algebras 13 References [1] L. C. Ciungu, Algebras on subintervals of pseudo-hoops, Fuzzy Sets and Systems 160(009, [] B. V. Gasse, G. Deschrijver, C. Cornelis, E. E. Kerre, Filters of residuated lattices and triangle algebras, Information Sciences (010 (online. [3] P. Hájek, Metamathematics of fuzzy logic, Kluwer, Dordrecht(1998. [4] L. Z. Liu, K. T. Li, Fuzzy filters of BL-algebras, Information Sciences 173( [5] L. Z. Liu, K. T. Li, Fuzzy Boolean and positive implicative filters of BL-algebras, Fuzzy Sets and Systems 15( [6] J. Mertanen, E. Turunen, States on semi-divisible generalized residuated lattices reduce to states on M V -algebras, Fuzzy Sets and Systems 159(008, [7] J. Rachuånek, D. Šalounová, Classes of filters in generalizations of commutative fuzzy structures, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 48(009, [8] A. B. Saeid, S. Motamed, Normal Filters in BL-Algebras, World Applied Sciences Journal 7(009, [9] E. Turunen, Mathematics Behind Fuzzy Logic, Physica-Verlag, Heidelberg(1999. [10] X. H. Zhang, Fuzzy logic and its algebric analysis, Science Press, Beijing(008. [11] X. H. Zhang, W. H. Li, On pseudo-bl algebras and BCC-algebras, Soft Comput 10(006, [1] X. H. Zhang, X. S. Fan, Pseudo-BL algebras and pseudo-effect algebras, Fuzzy sets and systems 159(008, No. 1,

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