ISSN 0304-9892
Jiianabha $110lllr
VOLUME37
Special Volume to Honour Professor S.P. Singh on his 70th Birthdav ..,
2007
Published by :
The Viji1ina Parishad of India DAYANAND VEDIC POSTGRADUATE COLLEGE
(Bundelkhand University) ORAI, U.P, INDIA
H. M. Srivastava University of Victoria Victorria, B.C., Canada
JNANABHA
EDITORS
AND R.C. Singh Chandel
D. V Postgraduate College Orai, U.P., India
Editorial Advisory Board R.G. Buschman (Langlois, OR) A. Carbone (Rende, Italy) K.L. Chung (Stanford) Pranesh Kumar (Prince George, BC, Canada)
L.S. Kothari (Delhi) I. Massabo (Rende, Italy) B.E. Rhoades (Bloomington, IN) R.C. Mehrotra (Jaipur) D. Roux (Milano, Italy) C. Pra._"<id (Aflahabad) V.K. Verma(Delhi) S.P. Singh (St. John's) K. N. Srivastava (Bhopal) L. J. Slater (Cambridge, UK.) H. K. Srivastava (Lucknow) R.P. Singh (Petersburg) M.R. Singh (London, ON, Canada) P.W Karlsson (Lyngby, Denmark) S. Owa (Osaka, Japan) N. E. Cho (Pusan, Korea)
Vijiiiina Parishad of India (Society for Applications of ~fathematics)
(Registered under the Societies Regi..c::tration ~~t XXI of 1860) Office: D.V. Postgraduate College, Orai-285001, U.P., India
President Vice-Presidents
Secretary-Treasurer Foreign Secretary
R.D. Agrawal (Vidisha) Karmeshu (Delhi) P.L. Sachdeva (Banglore) M.N. Mehta (Surat) AP. Singh (Jammu) Madhu Jain (Agra)
COUNCIL V. P. Saxena{Gwa1£or) G.C.Shanna(Agra) S.L. Singh (Hardwar) Principal D.V. Postgraduate College, (Orai) (N. D. Samadhia) R. C. Singh Chandel (Orai) H.M. Srivastava (Victoria)
MEMBERS P. Chaturani (Mumbai) A. P. Dwivedi (Kanpur) K. C. Prasad (Ranchi) S.N. Pandey ( Gorakhpur) KR. Pardasani (Bhopal) Ab ha Tenguria (Bhopal)
Jnanabha, 37, 2007 (Dedicated to Honour Professor S.P Singh on his 701
h Birthday)
PROFESSOR S.P. SINGH: AS A MA..."N" AND MATHEMATICIAl'T By
R.C. Singh Chandel Secretfil)~ Vijnana Parishad of India,
D.V Postgraduate College, Orai-285001, U.P, India
Professor S.P Singh needs no introduction. He is a remarkable man, a towering mathematician, a major topologist and well known eminent figure of non-linear Analysis: Approximation Theory and Fixed Point Theory. We feel great honoured to introduce Professor S.P Singh, who is very well actively associated with Jnanabha family since 1980. He was honoured by "Distinguished Service Award" of "Vijnana Parishad of India" in Silver Jubilee Conference of the Parishad held at Parishad Head Quarters : D. V. Postgraduate College, Orai, UP .. India in May 1996, for his distinguished services rendered to Mathematics and Vij nana Pa_risahd of India.
It is modest tribute at the occasion of his 70th Birthday Celeberations, when he has been as Honorary Fellow ofVij nana Parishad ofindia CF.VPI.)
during 12th Annual Conference of VPI held at J.N.V. University, Jodhpur (October 25-27, 2007).
Professor Singh, the ocean of generosity has inspired and helped many students, teachers and collegues to accept multidimensional challenges of physical realty.
PROFESSOR S.P. SINGH WITH HIS FAl~flLY MEMBERS Ancestral (Native) Place : Gopiganj, Varanasi, Uttar Pradesh, India (Exact Place of Birth) Date of Birth Father Mother Brother
Wife Son Daughter-in-law Daughter Son-in-law Grand Son Grand Daughter
Jan 27, 1937 : Late Sri Mahadeo Singh : Late Smt. Dharh1a Devi : Brahm Din Singh
C ma Shankar Singh Late Professor KL. Singh (Univ. of Fayetteville State Cniversity; N.C. U.S.A.) Smt. Suman Singh Manoj Singh Tanuja (D/o Dr. Panjab Singh Former V.C., B.H.U.) Indu Sudha (Anesthsia) Abhay Kumar (Psychiatric) Shivank (S/o Manoj) Priyanka (D/o Indu Sudha)
4
Full Name Address
PROFESSORS. P. SINGH AT A GLANCE : S~atha Prasad Singh : 750 Hedgerow Place, London, ON, Canada N6G5H9 : Physics Department University of \Vestern Ontario,
London, Canada, N6G3K7 Telephone (519) 6612111 Ext. 86455 (Office)
(519) 6727540 (Home) Fax: (519) 6612033
Email : [email protected] [email protected] [email protected] [email protected]
Qualifications B.Sc. : 1957(Agra University), Agra, India) M.Sc.: Ph.D.:
1959 (Banaras Hindu University), Varanasi, India) 1964 (Banaras Hindu University, Varanasi, India)
Teaching Experience Banaras Hindu University Univers,; v of Illinois, USA Wayne state University, USA University of Windsor, Canada Memorial University, Canada Memorial University, Candad . Shawnee State University, USA
Lecturer Lecturer Asstt. Professor Asstt. Professor Assoc. Professor Professor Adjunct Professor
1959-63 1963-64 1964-65 1965-67 1967-72 1972-2002 (shorter vi.sit-a few times:
University of Western Ontario, Visiting Professor 2002-continued Membership American Math. Soc, Canadian Math. Soc, Allahabad Math. Soc., Indian Math. Soc., Ins., of M&th. & Its Applications, Vijnana Parishad of India, Bharat Ganita Parishad, Poorvanchal Academy of Sciences, Unione Mat Italiana, Indian Academy of Mathmatics, Soc. of Special Functions & Applications. Fellowship Fellow of the Inst. of Math. & Its Applications, (F!MA (UK))
Fellowship of the National Academy of Sciences, India (FN.A Sc)
Fellowship of the Allahabad Math. Society (FALMS) Fellowship of Vijnana Parishad India (FVPJ)
Editor Bulletin, Vijnana Parishad of India Foreigp Edito~ Journal ·. f Natural and Physical Sciences
Member Council of the Allahabad Mathematical Society Member Editorial Board
5
Ganita (Bharat Ganita Parishad), Poorvanchal Academy of Science, Jnanabha
(Journal 1Vijnana Parishad of India), Epsilon Journal of Mathematics, Far East J. of Math Sci., Intern. J. of Sci. & Eng., Journal of Inequalities in Pure and Applied
Math (JIPA.J.W), Lecture Notes: Allahabad Math Society, Vikram J. of Mathematics,
Varahmihir J. of Math Sci., Nigerian J. of Math Sci., Southeastern Asian J of Math & Math Sciences, Australian J. of Math Analysis & Appl. (AJMAA), Oriental J. of Mathematics, Global J. of Math & Math Sc. (GJMMS), Intern. J. of Pure & Appl. Math Sciences (lJPA.J.'v!S), Intern. J. of Math Sciences., Chhatisgarh J. of
Mathematics and engineering, . Journal of Fixed point Theory
Member Advisory Board i) Indian Academy of Mathematics
Nigerian Journal of Scientific Research International J. of Special Functions and Applications
Member Research Board of Advisors The American Biographical Institute Reviewer Mathmatical Reviews, USA; Zentrablatt fur Mathematik, Germany
Director-Nato-Advanced Study Institute i) Approximation Theory - 1982-83
ii) Nonlinear Functional Analysis - 1984-85
Approximation Theory, Spline Functions & Applications- 1990-91
iv) Approximation Theory; Wavelets & Applications - 1994-95
Project Leader NATO- International Research Group - 1980-R2.
Research Field Approximation Theory, Fixed point Theory (Nonlinear Analysis), Variational
Inequalities, Special Functions, Mathematical Economics. Research, in part, has been supported by NSERC (Natural Sciences and
Engineering Research Council, Canada) NATO (North Atlantic Treaty
Oraganization), CNR (Italian Research Council), Canadian Math Congress,
University Grants Commission, India and Memorial University, Secretary of State,
Government of Canada.
Books and Proceedings 1. Lecture 11\/otes on Fixed-point Theorems in Metric and Branch Spaces,
Matscience, Madras (1974), p. 112.
6
2. Proc. Intern. Conf on Nonlinear Analysis & Applications, Marcel Dekker, New York (1982). p. 465 (with J. Burry).
3. Proc. Conf on Topological Methods in Nonlinear Analysis, Contemp. Math., American Mathl. Soc. (1983), p. 218 (with S. Thomeier and B. Watson}.
4. Proc. NATO-AS! on Approximation Theory & Spline Functions, D. Reidel Publ. Co. (1984), p. 485(with J. Burry and B. Watson).
5. Proc. Intern. Conf on Approximation Theory, Pitman Publ. Co., London, UK (191~5), P. 264.
6. Proc. NATO-AS! on Nonlinear Functional Analysis, D. Reidel Publ. Co. (1986}, p. 418.
7. Proc. Intern. Conf on Solutions of Operator Equations & Fixed Points, The Math. Soc. Res. Inst. Korea (1986), p. 250 (with V.M. Seghal & J. Burry).
8. Banach Contraction Principle and its Applications to Integral Equations, Memorial University of New Foundland, C-CORE (1988), p. 42 {w>ith J. Walsh).
9. Proc. NATO-AS! on Approximation Theory, Spline Functions and Applications, Kluwer Academic Publishers (1992), P 475 (with A. Carbone, R.
Charron, and B. Watson).
10. Proc. NATO-AS! on Approx. Theory, Wavelets and Applications, Kluwe'r Acac'emic Publishers (1995), p. 572(with A. Carbone and B. Watson).
11. Fixed point Theory and Best Approximation: The KEJvi-Map r11nc'iJM1:;.
Academic Publishers (1997), p. 220 (with B. Warson and P Sri\1astava}r.
12. Proc. International Con{. on Nonlinear Nonlinear Analysis Forum 6 (2001) p. 275
External Lectures 2006: Banaras Hindu University
and Its
4 lectures (Science College, Math. Engineering College,
Faculty of Management, and Women's College) Math. University of Allahabad
C? Open University Allahabad.
O~ganized a Special Session in the American Math Society a.nd presented a paper. October, 2006, Cincinnati, USA
2005 : HRI, India
IIT Roorkee, India,
Intern. Conf. Cetraro, Italy 2004 : University of Calabira, Italy
University of Perugia, Iniversity of Roma 3, ICTP, Trieste Italy
2003 : Fniversity of Perugia, Italy
University of Calabria, Italy 2002: Aligarh Muslim University India
Indian Math. Society Meeting, Jan. 27-31. Carleton University, April 5 University of Calabria, Math Dept. June 18 Univ. of Calabria, Centre of Num. Analysis, June 19, Italy
7
2001 University of Calabria, Italy International Conference on Functional Analysis Methods in Economics and Finance, Italy, June 28-July 2, 2001.
2000 University of Calabria, Italy University of Rome, Italy University of Torino, Italy University of Florence, Italy University of Siena, Italy University of Perugia, Italy
1999 Shavvnee State University, Ohio, USA University of Calabria University of Torino
1998 Sha¥ln.ee Stace University (4 lectures) University of Torino Italy (2 lectures) University of Bari, Italy University of Calabria, Italy (2 lectures)
1997 University of Rome, Italy University of Bari, Italy University of Calabria, Italy University of Torino, Italy American Math Society Meeting, Detroit R.D. University, Jabalpur, India Allahabad Math Society
1996 University of Seina, Italy University of Perugia, Italy University of Salerrno, Italy University of Rome, Italy University of Torino, Italy (Series of 4 lectures) Silver Jubilee Conference, Vij nana Parishad of India Delhi University, Delhi, India
1995 FAMA=95, Cosenza, Italy University of Calabria, Italy University of Perugia, Italy University of L=Aquila, Italy
8
Italian Research Council (CNR), Naples University of Catania, Italy Banaras Hindu University, India Mehta Research Institute, Allahabad, India Delhi University, Delhi, India IIT Delhi, India Panjab University, Chandigarh Gurukul Kangari University, Hardwar The National Seminar in Math, Bhopal A '_)proximation Theory Conference, Taxas
1994 KATO-ASI, Maratea, Italy University of Calabria, Italy University of Torino, Itlay (2 lectures) University of Perugia, Italy University of Rome, Italy Royale Military College, Kingston, Ontario {Colloquium)
1993 Conf. del Seminario Mat., University of Torino, Italy University of Torino, Italy Universita di Calabria, Italy University of Firenze, Italy Lniversity of Rome II, Rome, Italy University of Rome, Italy Universita di Calabria, Italy Dalian Institute of Technology, China 1 Series 4 lectures) Beijing University, China
1992 Lunknow University, India Allahabad Math Society, India Udai Pratap College, India Banaras Hindu University, India University of Torino, Italy (Series of 4 lectures) ·e:iiversity of Calabria, Italy L niversity of Napoli, Italy University of Rome, Italy
1991 University of Bari, Italy Italian Research Council (CNR), Napoli, Italy University of Milano, Italy University of Florence, Italy
1990 Banaras Hindu University, India University of Messina, Italy
University of Calabria, Italy Scuola Intern. Superiore di Studi Avanzati (SISSA), Trieste, Italy International Centre of Theoretical Physics (ICTP), Trieste, Italy
1989 Allahabad Math Society, India Banaras Hindu University, India University of Calabria, Italy
1988 Fayetteville State University, North Carolina, USA Carleton University, Ottawa, Ontario Laval University, Quebec City, Quebec Lakehead University, Thunder Bay, Ontario Concordia University, Montreal, Quebec Concordia University, Montreal, Quebec University of Milano, Italy University of Calabria, Italy University of Napoli, Italy Universicy of Catania, Ital.y Universicy of Delhi, India UT Bombay, India {series of 4 lectures) Mehta Research Institute, Allahabad, India (series of 3 lectures)
9
1987 University of Prince Edward Island, Charlottetown, Prince Edward Island University of Milano, Italy Universicy of Calabria, Italy University of Napoli, Italy UT Bombay, India Mehta Tesearch Institute, Allahabad, India
Conferences/Meetings Attended 2005 Nonlinear functional analysis applications in economics and Finance,
Cetraro , Italy 2003 ISAAC Conference, Toronto 2002
" 1. Indian Mathematical Soc. Meeting, Aligarh, India January 27-31, 2. Vijnana Parishad of India (VPI) Meeting, Delhi, India Feb. 24-26 3; D.S.T meeting, Allahabad University, Feb. 12-14
2001 International Conference on Functional Analysis Mathods in Economics and Finance
2000 1. Organized a Special Session in Toronto AMS Meeting, September 2000. 2. Presented a paper in Toronto Meeting. 3. Att.ended Allahad Math. Soc. Council Meeting.
10
4. Attended AMS Annual Meeting, Washington, DC, January· 2000.
1999 Organized a conference on Nonlinear Analysis and Applications, Canadian Mathematical Society Summer Meeting, May.
1998 American Mathematical Society, Louisville, Kentucky March.
1997 A..merican Mathematical Society, Detroit, April.
1995 , 1 sth Texas Conference on Approximation Theory, January 7-12.
2. National Seminar in Mathematics, Bhopal, India, March 13-14.
3. International Conference on Functional Analysis, Methods and Applications, Cosenza, Italy, May 27-June 2.
1994 NATO-ASI on Approximation Theory, Wavelets an.d Applications, Italy, May 16-26.
1993 1. Conference on Functional A_nalysis and Applications, Milano, Italy, May, 10-14.
2. Topological Analysis Workshop on Degree Theory, Singularity and Variations, Rome, May 28-31.
3 NATO-ASI, Montreal, July 26-August 6.
4. NATO-ASI, Funchal, Madeira, Portugal, August 6-21.
1992 Allahabad Math Society, February 7-8.
1991 1. NATO-ASI, Maratea, Italy, April 28-May 2. Canadian Mathematical Society Summer snPrnrrM"Alit"P
29-June 02. 3. Research Workshop on
Sherbrooke, May 28-June
Variational .Methods,
4. Second International Conference on Fixed point Theeory, Halifax, June 9-14.
1990 1 NATO-ASI on Shape Optimization and Free Boundaries Montreal,
June 27-July 13.
2. Conference on Recent Developments in Analysis, Manifolds and
Application, Banaras Mathematical Society, Varanasi, India, Januar:;;r 3-5.
3. 1989 1.
2.
3.
4.
Ontario Mathematical Meeting, St. Catherines, April 18-20
Indian Mathematical Society Annual Meeting, Delhi, December 26-30.
International Conference on Approximation Theory, Jahalpur, India, December 15-18.
APICS Meeting, Sackville, New Brunsv.ri.ck, October 27-28.
International Conference on Functional .A...rialysis, Cetraro, Italy, June 26-30.
11
5. International Conference on Fixed Point Theory, Marseille, France,
June 5-9.
{.
8. 1988 1.
2.
1987 l.
2. 3.
NATO-ASI, Columbus, Ohio, May 21-June 3. American Mathematical Society Meeting, Chicago, Illinois, May 19-
20
~>\llahabad Mathematical Society, Allahabad, January 20. American Mathematical Society Meeting, Atlanta, January 12. NATO-ASL Istanbul, Turkey, August 15-18. Canadian Mathematical Society Meeting, Kingston, Ontario, June. Learned Societies Meetings, Hamilton, Ontario, June. American Mathematical Society Meeting, Salt Lake City, Utah, (chaired one session), August.
4. Allahabad Mathematical Society, November.
Publications 1. S.P. Singh, K Tarafdar and B. Watson, A generalized Urysohn imbedding and
,._,.,.,..,.,,rr fixed point theorem in topological space, Indian J Pure & Appl.
Math., 34 227-673.
2. E. Tarafdar, RP. Singh ai.~d B. Watson, Fixed point theorems for some extensions of contraction mappings in uniform spaces, J. Math Sci. l (2002) 53-61.
3. J. Li and S.P. Singh, An extension of Ky Fan=s best approximation theorem. Nonl. Anal. Forum, 6 (2001), 163-170.
4. M. S. R. Chowdhury, E. Tarafdar, S.P. Singh and B. Watson, Generalized quasivariational inequalities for Hemi- continuous operators on non-compact sets, Nonl. Anal. Forum 6 (2001), 79-90.
5. P. Kumar, S.P. Singh, and S. S. Dragon1ir, Some inequalitties involving beta and gamma functions, . A_.71,af. Forum 6 {2001), 143-150.
6. G. Isac, \~ M. Sehgal and S. P. Singh, An alternate version of a variational _ _ J Ma.th., 41 1999), 25-31.
7. S. P. Singh, E. Tarafdar and B. Watson, A generalized fixed point theorem and equilibrium point of an abstract economy, J. Comp. Appl. Math., 113 (2000), 65-71.
8. T. D. Narang and S. P. Singh, Best coapproximation in metric linear spaces, Tamkang J lvfath. 80 (1999), 241-252.
9. J. Li and S. P. Singh, An extension of Fan=s theorem, G.K. Vignana Patrika 1 (1998), 133-146.
S. P. Singh, Best approximation and fixed point theorems, Proc. Nat. Acad. Sci. India, LXVII (1997), 1-27 (survey paper).
11. S. P. Singh, E. Tarafdar and B. Watson, Variational inequalities and applications, Indian J Pure Applied Math. 28 (1997), 1083-1089.
12
12. T. D. Narang and S. P. Singh, Best coapproximation in locally convex spaces,
Tamkang J. Math. 28 (1997), 1-5. 13. S. P. Singh, E. Taraf dar and B. Watson, Variational inequalities for a pair of
Pseudomonotone functions, Far East J. Math Sci. Special Volume (1996), 31-
52. 14. V M. Sehgal and S. P. Singh, Coincidence thorems for paracompact sets, J
Indian Math Soc., 64 (1997), 97-101. 15. D. V Pai and S. P. Singh, On p-centers and fixed points, Proc. Intern. Conf on
Approximation Theory and Its Applications, New AGE Intern. Publishers, Ed. G. S. Rao (1996), 87-98.
16. A. Carbone and S. P. Singh. On Fan=s best approximation and applications, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996J, 35-52.
17. V M. Sehgal and S. P. Singh. The best approximation and an e:x'tension of a fixed point theorem ofF E. Browder, Intern. J }lfath and .-,.,foth. Seo., 18 (1995),
745-748. 18. S. P. Singh, On Ky Fan=s best approximation and fixed point theorems, J.
Nat. & Physical Sci., 9-10 (1995-96), 25-42.
19. S. P. Singh, On Fan=s best approximation theorem, Proc. Intern. LXJnf. on
Approximation Theory VII, World Scientific Puhl. Co. Eds. Chui and Schmnaker (1995), 529-536.
20. S. Park, S. P. Singh and B. Watson, Some fixed romposnes
of acyclic. maps., Proc. Amer. Soc .. , 25 158. 21. S. Park, S. P. Singh and B. Watson, Remarks on best approximations and fixed
points, Indian J. Pure Appl. Math Soc.,, 25 (19941, 459-462.
22. S. P. Singh and B. Watson, Best approximation and fixed point theorems, Proc. NATO-AS!, Kluwer Academic Publishers (1994), 285-294.
23. G. Marino, P. Pietramala and S. P. Singh, Convergence of approximating fixed
points sets. for multivalued mappings, J Math. Sci., 28 (1994), 117-130.
24. A. Carbone and S. P. Singh, On some fixed point theorems in Hilbert spares, Dem. Math. XXVI (1993), 649-656.
25. S. F. Singh and B. Watson, On convergence results in fixed point theory, Rend. Sem. Math. Univ. Pol. Torino, 51 (1993), 73-91.
26. S. P. Singh and S. Sessa, Application of the KKIVI-principle to Prolla type
theorems, Tamkang J. Math. 23 (1992), 279-287.
27. V M. Sehgal and S.P. Singh, On best simultaneous approximations in normed
linear spaces. Proc. NATO-AS!, Kluwer Academic Publishers (1992), 463-4 70.
28. V M. Sehgal and S.P. Singh and B. Watson, Best approximation and fr;{ed points, Bull. Allahabad Math. Soc., 6 (1991), 11-27.
29. V M. Sehgal and S.P. Singh and G. C. Gastl, On a fixed point theorem for
13
multivalued Maps, Proc. Intern. Conf On Fixed Point Theory and Applications, Eds. Baillon and Thera, Pittman Publishers (1991), 377-382. S. P. Singh and P V Subrahmanyam, A note on fixed points and coincidences. J Poorvanchal A.cad. Sci., 2 {1991), 1-5.
31. V.: M. Sehgal, and S. P Singh, A Ky Fan type theorem and its applications, Progr. Math., 24 (1990), 13-18.
32. V.: M. Sehgal, S. P Singh and J. H. M. Whitfield, KKM-maps and fixed point theorems. Indian J. }vfath., 32 {1990), 289-296.
33. S. Sessa a11d S. P Singh, Applications of KKM-Map principle, J Pooruanchal Acad. Sci., 1 0990), 1-6.
34. V: M. Sehgal and S.P Singh, A theorem on best approximation, Numer. Funct. Anal. Optimiz. IO (1989), 181-184.
35. A. Carbone, B. E. Rhoades and S.P Singh, A Fixed point theorem for
generalized contraction map, Indian J Pure Appl. Math., 20 (1989), 543-548.
36. D. Roux and S. P. Singh, On some fixed point theorems, Intern J. Math & Math Sci. 12 (1989), 61-64.
37. V. M. Sehgal and S. P Singh, Fixed point theorem for contraction type mappings in convex spaces, Jlianahha, 19 (1989), 11-14.
38. D. Rou...'!C and S. P. Singh, On a best approximation theorem, Jnanabha, 19 (1989), 1-9.
39. V. M. Sehgal and S.P Singh, A generalization to multifundions of Fan=s best approximation theorem, Proc. Amer. Math. Soc., 102 (1988), 534-537.
40. S.P Singh, On Ky Fan=s theorem and its applications 1, Rend. Sem. Math. &
Fis. di 41. A. Carbone and S. P Singh. Fixed point theorems for Altman type mappings.
Indian J. Pure Jfaih. 18 ;1987), 1082-1087.
42. V: M. Sehgal a.11d S.P. Singh ai1d RE. Smithson, Nearest ponts and some fixed point theorems for weakly compact sets, J. i\1ath. Anal. Appl., 128 (1987),
108-111.
43. S. Massa, D. Roux and S. P Singh, Fixed point theorems for multifuctions, fodian J Pure Appl. lvfoth. 18 \1987), 512-514.
44. S. P Singh, On Ky Fan =s theorem and its applications II,. Nonlinear Analysis (Ed. T.M. Rassias). World Sci. Publ. Co. (1987), 527-37.
45. V M. Sehgal and S.P Singh, On rendom approximation and a random fixed point theorem, MSRI Korea Publ., 1 (1986), 99-102.
46. S. P Singh and B. Watson, On approximating fixed points, Proc. Symp. Pure Math., Amer. Math. Soc., 45 (1986), 393-395.
47. S. P. Singh, S. Massa and D. Roux, Approximation techniques in fixed point
theory, Rend. Sem. lviat. Fis. di Milano, LIII(l983), 165-172.
14
48. V M. Sehgal and S.P. Singh, On rendom approximation and a random fixed point theorem for set-valued mappings, Proc. Amer. lvlath. Soc., 95 (1985), 91-
94. 49. V M. Sehgal and S.P. Singh, A theorem on the minimization of condensing
multifunction and fixed points. J. Math. Anal. Appl., 107(1985), 96-102. 50. H. M. Srivastava and S. P. Singh, A note on the Widder transform related to
the Poisson integral for a half plane. Inter. J Math. Edu. Sci. Tech. 16 (1985),
675-377. 51. V M. Sehgal and S.P. Singh, A Variant of a theorem of Ky Fan, Indian J. Math.
Soc., 25 (1984), 171-174 .. 52. V M. Sehgal and S.P. Singh, A fixed point theorem :in reflexive Banach spaces,
Math. Sem. Notes, 11 (1983), 81-82. 53. S. P. Singh and B. Watson, Proximity maps and fixed points, J. Approx. Theory,
39(1983), 72-76. 54. V M. Sehgal, S.P. Singh and B. Watson, A coincidence theorem for topological
vector spces, Indian J Pure Appl. lv!.ath., 14 (1983), 565-566. 55. S.P. Singh and B.N. Sahney, On best simultaneous approximation in Banach
spaces, J.Approx. Theory, 35 (1982), 222-224.
56. S. F'. Singh, K. M. Das and B. Watson, A note on Mann iteratirm for quasinonexpansive mappings, Nonlinear Anaiysi:s, Tr\!L4. 5 675-676.
57. S. P. Singh and B. N. Sahney, On best simultaneous approximation, Pni<:!:ei::tu.1
of the International Conference on (Edited by W Cheney) (1980), 783-789.
Academic press
58. S.P. Singh. Some results on best approximation ~n locally convex spaces, J. Approx. Theory, 28 (4)(1980), 329-332.
59. S. L. Singh and S.P. Singh, A fixed point theorem, lnd.JPure Appl. Math., 11 (1980) 1584-86.
60. S.P. Singh, An application of fixed point theorem in approximation theory. J. App: 'JX. Theory, 25 (1979), 89-90.
61. S. P Singh, Applications of fixed point theorems to approximation theory. Proc. Intern. Conf on Nonlinear Analysis, Ed. V Lakshmikantham, Academic press (1979). 389-391.
62. V. M. Sehgal and S. P. Singh, A fixed point theorem for the sum of two mappings, Math. Japan. 23 (1978), 71-75.
63. B.K. Ray and S. P. Singh, Fixed point theorems in Banach spaces, Indian J Pure & Appl. Math. 19 (1978), 216-221.
64. B. Meade and S. P. Singh, An extension of a fixed point theorem, Math. Semi. Notes, Japa,n, 5 (1977), 443-446.
65. S. P. Bingh and B. Meade, On common fixed point theorems, Bull. Au.str. Math.
15
Soc. 16 1977:. 49-53. YIR55 # 11234.
66. S . .P Singh and V M. Sehgal. Some fixed point theorems in metric spaces, J (1976), 35-44.
67. S. P Singh, On common fixed points for mappings, Math. Japonicae, 21(1976),
283-264. Mr55#1343. 68. V M. Sehgal and S. P Singh, On a fixed point theorem of Krasnoselskii for
com:ex spaces. Paci{ J. Jfoth., 62(1976), 561-567. MR54 # 1032.
69. S. Deb and S. P Singh, Fixed point theorems in topological vector spaces, Ann. Soc. Sci., Bruxelles 88 \ 1974), 273-280. MR 50 # 5564.
70. S. P. Singh, On the convergence of the sequence of iterates, Atti. Acad. Naz. Lincei, LVH <1974), 502-505. MR55 # 8908.
71. S. P. Singh. Densifyingmappings in metric spaces, .\lath. Student, XLI, 4(1973),
433-436. MR52 # 9206.
72. S.P. Singh and R K. Yadav, On the convergence of sequence of iterates, Ann.
74_
Soc. Sci. 87 1973', 279-284. MR 48 # 12177.
S.P Singh, Fixed point theorems for a sum of nonlinear operators, Atti. Acad.
Naz. Lincei .. 54 fl973\. 558-561.
theorems of Reinermann, Atti. Acad.l 1\Taz. Lincei, 54
75. S. P Singh, Fixed point theorems in metric spaces, Riu. 1972), 37-40. MR 51 # 4203. 279-282. MR54 # 1033.
Unit-. Panna
76. L. S. Dube and S. P. Singh, On sequence of mappings and fixed points, Nanta Math. 5 ( 1972), 84-89. MR47 # 9597.
77. S. P Singh and .M. Riggio, On a theorem of Diaz & Metcalf, Acad. Naz. de
lincei, 13 •• 1972 84-89. MR49 # 3615. 78. S. P Simrh and B. Holden. Some fr'(ed point theorems, Bull. Soc. Royal Liege
79. S. P. Singh and F Zorzitto, On fixed point theorems in metric space, Ann. Aoc., Sci ... Bruxelles, 85 (197li. 117-123. MR 46 # 2661.
80. S. P Singh, On fixed points, Publ. lvlath. Inst. 11(25) (1971), 29-32. MR46 #
81. S. P. Singh and C. Norris. Fixed point theorems in generalized metric spaces,
Bull. Math. Roumania, 14 i.l970l, 1-5. MR49 # 1503.
82. L. S. Dube and S. P Singh, Tvm theorems on fixed points, Banaras Math. Soc. J., 2 1969l. 5-7.
83. L. S. Dube and S. P Singh, On multi-valued contraction mappings. Bull. Math. Roumania, 14 (1970),308-310.
84. S. P Singh, On a fixed point theorem in metric space, Rend. Sem. Mat. Padoua, 18 (1970), 229-232. MR 43 # 5517.
85. S. P Singh, On a sequence of contraction mappings, Riv. Mat. Univ. Parma, 11
16
(1970), 227-231.
86. S. P Singh, Some theorems on fixed points, Yokohama 25. MR 43 # 4016.
J 18 1970L. 23-
87. S. P Singh, Some results on fixed point theorems, Yokoham.a Afoth. J, 17 ll969), 61-64. MR 41 # 2488.
88. S. P Singh, On commuting functions and fn::ed points, Riv. 1vlat. 10 (1969), 91-94. MR 45 # 1146.
89. S. P Singh, On a theorem of Sonnenschein, Bull. Acad. Ro~vale Belgique, 55 (1969), 413-414. MR 41 # 5516.
90. S.P Singh, Sequence of mappings and fixed points, Ann. Soc. Sci. Bruxelles, 83 J969l, 197-201. MR 41 # 2638.
91. S. P Singh and W Russell, On a sequence of comraction mappings, Canan'. Math. Bulletin 12 (1969) 513-516. MR 40 # 2049.
92. S. P Singh, On a fixed point theorem in metric space, Delta 1(4) 1968;70). MR 42 # 847.
93. S. P Singh, A note on K-transforms, Riv. Mat. Univ. Parma, 9
94. S. P. Singh and KL. Singh, On fixed point theorems, (1967), 385-389. MR 40.
95. S.P. Singh, A note on a fixed point theorem, Riv. 161-168. MR 40 # 3532.
96. S.P. Singh, A theorem on generalized B.H."U, 17 (1966/67), 154-162 .. MR
97. S. P Singh, On the resulta.11t two kernels, 221-230.
98. S.P. Singh, Infinite integrals involving G-fun~crions. ( 1966), 225-235. MR37 # 5620.
J.
Sic .. 33 1967
Parma 7
99. S. P. Singh, Two theorems on generalized Meijer-Laplace transform, Riv. i\fat.
Univ. Parma, 7 (1966), 115-125. MR37 # 5619.
100. S. P. Singh, Two·theorems on generalized Hankel transform, Ganita 15 1964),
75-83. MR 35 # 5867.
101. S. P. Singh, A property of self-reciprocal functions, Gcmita, 15 (1964), 9-18. MR: '2 # 6162.
102. S. P. Singh, Some theorems on a generalized Hankel transform, Acad., Sci. India, Sect. A 34(1964), 5- 10. MR 36 # 4278.
103. S. P Singh, Two theorems on the generalized Handkel transform. Ganita, 15 (1964), 75-83. MR 35 # 5867.
104. S. P. Singh, A property of self-reciprocal functions, Ganita, 15 1964), 9-
MR 32 # 6162. 105. S. P. Singh, On kernels in generalized Hankel transform, Proc. Nat. A.cad. Sci.
17
A 33 1963 395-406.
106. S. P Si..TJigh, A generalized Hankel transform, Proc. Nat. Acad. Sci. India, Sect. ,331 423-430.
107. S. P Singh Relations bet•veen Hankel transform and generalized
Laplecetransform, Proc. Xat. Acad. Sci. India, Sect. A, 33 (1963), 431-
108. S. P S1ngh, On the generalized Hankel transform, Proc. Nat. Acad. Sci. India,
Set. A, 33\ 19631, 327-332.
109. S. P Singh, Certain kernel functions, Proc. Nat. Acad. Sci. India, Sect. A 33
1963;, 413-422.
1 S. P Singh. Certain kernels in generalized Hankel transform, Proc. Nat. Acad.
Sci. India, Sect. A 33 (19631, 337-348.
111. S. P Singh. Some kernel functions. Proc. Nat. Acad. Sci. India, Sect. A 33 1963!, 321-326.
112. S. P Singh. Ge!1eralized Hankel transform and self-reciprocal functions, Proc.
Sect. A 33 1963), 309-316.
113.S. P Singh, On kernel functions, Ganita, 14 (1963), 105-113. MR 29 # 5068.
114. S.P Singh, On
14
Hankel transform and self-reciprocal functions,
22-42. ?\IR 29 # 5067.
115. S.P Singh. A property of self-reciprocal functions of two variables, Vijnana Parishad Amusandhan Patrika 5 (1962), 45-58. MR 27 # 1787.
116. S. P Singh, On Besselian functions, Vijnana Parishad Anusandhan Patrika 5 (19621, 21-27. 1vfR27 # 1634.
117. S. P. Singh, Relations between Hankel transform and the generalized Laplace
transforms CWnittaker & 1V1eijer transforms\ Proc.
ii, 32 (19£2 , 355-359. MR28 # 2412.
Acad. Sci. India, Sect.
118. S. P Singh, Some properties of a generalized Laplace transform, J. Sci. Res.
Banaras Cnir..' .. 13, 1962 63 '· 344-360. MR 29 # 2604.
119. S.P Singh, Certain
1962/63), 252-266. J. Sci. Res. Banaras Hindu Uniu., 13
120. S. P Singh, Self-reciprocal functions of two variables in the generalized Hankel
transform, J. Sci. Res. Banaras Hindu University, 13 (1962/63), 182-191, MR
27 # 2815 c.
121. S. P Singh, A property of self-reciprocal functions, J. Sci. Res. Banaras Hindu
University 13 (1962/63), 18-23. Iv1R 27 # 2815 b.
122. S.P Singh, Kernels in the generalized Hankel transforms, J. Sci. Res. Banaras
Hindu Unil:ersity, 13 (1962/63), 9-17. MR27 # 2815 a.
123. S.P Singh, On certain properties of generalized Hankel transforms, Jour.
Indian .lvfath. Soc. 26 (1962), 35-52. MR26 # 4128.
124. S.P Singh, Certain recurrence relations, Ganita, 13(1962), 1-8. MR 29 # 6076.
18
125. S.P Singh, Certain recurrence relations involving a generalized function of the Bessel type, Jour. Sci. Res. Banaras Hindu Uniuersity, 12(1962), 392-400 ..
MR ~27 # 384 7. Service to .Memorial University
University Committees 2002 MUNFA pension com., .\fCNFA scolarship com., Board of Regems
pension Com.
2001
2000
1999
1998.
1997.
1996.
1995.
1994.
1992.
1991.
1990.
MUNFA pension, MUNFA scholarship, Allahabad Math. Soc. CounciL Sentate Advisory Commitee of University Library, Board of regents
pension commitee Senate Committee on· Scolarships and Financial Aids, .MlJNFA pension Committee, Scholarship, Allahabad ?\fathmaric>l': Society Council, Senate Advisory Commitee of University Libary Senate Commitee on Copyright, Senate Committee on Scholarships and Financial Aids, Senate Ad\isory Commin:ee of University Libarary, Board of Regents Pension Committee, MUNFA, Pension, Scolarships, Allahabad
Mathematical Society Council Senate Commitee on Copyright, Senate Committee on Scholarships Financial Aids, MUNFA, Pension, Scholarhips, }Jlahabad Mathemat1caI Society Council. Senate Committee on CopyTight, Senate
Financial Aids, MUNEA, Pension 0'-'-"'~'""'u Society Council. Senate Committee on Cop:JTighL Senat,e on Scolarships and
Financial Aids, MUNFA, Pension, :Scholar:ships. Allahabad ~\fathematical Society Council
Senate Committee on CopyTight, Senate Committee on Scholarships and Financial Aids, MUNFA, Scholarships, Allahabad Mathematical Society Council. Senate Committee on Copyright, MUNFA, Scholarships, Pension,
Allahabad Mathematical Society Council.
Academic Council of Graduate Studies, MUNFA Communications. Scholarship, Senate Advisory Committee on Library, Shastri IndoCanadian Institute. Academic Council of Graduate Studies, MUNFA Communications, Scholarship, Senate Advisory Committee on Library, Shastri IndoCanadian Institute.
Academic Council of Graduate Studies, MUNFA Communications, Scholarship, Senate Advisory Committee on Library, Shastri Indo-
19
a..HauH:ui Institute.
1989. Academic Council of Graduate Studies, Senate Advisory Committee on
Shastri Indo-Canadia..ri Institute, NSERC screeing Committee.
1988. Academic Council of Graduate Studies, Senate Advisory Committee on
Library, Shastri Indo-Canadian Institute, NSERC screeing Committee.
Departmental Committees
1999 Graduate Studies Commitee
1998 . Graduate Studies Committee
Search Committee
Search Committee
1995 Search Committee
1992 Search Committee
1991 Promotion & Tenure Committee, Search Committee
1990 P1·omotion & Tenure Committee, Search Committee
1989 Graduate St-udies Committee, Library Committee
1988 Graduate Studies Committee, Library Committee
2001 s w F
2000 w F
1999 \V
F
1998 s F
1997 F
s 1996 F
s i995 F
w 1994 w
s F
1993 w 1992 F
s 1991 F
Sabbatical
Math 1090
Sabbatical
TE..\CHING RECORD
Math 2000, 3260
Math 1001. 1081
2000, 3202
Math 2000,
Math 2000, 3260
Math 32 3301 Sabbatical
Math 2000, 3260
Math 1080,1001
Sabbatical
Math 1081. 4310
Math 1080, 3210
Math 2000, 3260
Math 1081, 3201
20
W Math 1081, 3210
1990 F w
1989 F
Math 1081,2001 Math 1080 ,2000 Math 1001, plus labs
THESES SUPERVISED MASTER'S THESES C.W Norris
W Russell L. S. Dube PS. Cheema
Fixed point theorems in generalized metric spaces and applications
Fixed point theorems in uniform spaces Nonexpansive mappings and applications Fixed point theorems in metric spaces and applications
F. Hynes Iterated contraction mappings and applications W I vimey Multivalued contraction mappings B. Holden Sum of nonlinear operators and fixed puun,:,
M. I. Riggio Measure of noncompactness, densifying mappings and fixed points
S. Deb Quasibounded mappings, fixed point theorems and applications
R. K. Ya fav GRometry of Banach spaces and some fixed point theorems
M. Veitcl1 Fixed point theorems for mappings with a convexity
G. Collins Extension of some fixed point theorems in metric spaces A. Robertson Convergence of the sequence of successive approximations B. Meade Extension of some fn:.ed poim theorems and aoouranoins HONOR'S THESES R. Blair D. Payne
D. Ryan
Banach contraction
Conformal mappings
Contraction mapping
ACKNO'\\'LEDGE::\IENT I wish to record my deepest and sir.cerest feelings of gratitudes to
Smt. Su1r:an, the wife of Professor S.P Singh for helping me in collecting the data mentioned above.
Jnanabha. . 2001 !Dedicated to Honour Professor S.P Singh on his 701
h Birthday)
A CORRECTION TO A.'t EXA..:."1PLE OF RENU CHUG A1'.TD SANJAY h.'U'1dAR FOR WEAKLY CO:\IPATIBLE SELF-MAPS
B\-T. Phaneedra
Department :\fathematics_ Hyderabad Institute of Technology and
:Management (HITAJV1\ Gov.-davelli Village, Medchal Mandal, R.R. Distt.,
Hyderabad, A ... -ridra Pradesh, India
E-mail : tp_indra\[r rediffmail.com
rReceiued : February 15, 2007;
ABSTRA.CT A correction is suggested in an example of Renu Chug and Sanjay Kumar,
in attempting to disprove the converse of the statement that every compatible
pair is weakly compatible.
2000 Mathematics Subject Classification : 54H25
Keywords and Phrases : Weakly compatible self-maps. 1. Introduction. Let 1X,d; denotes a metric space. As a generalization of
commuting self-maps, Gerald Jungck [2] defined self-maps S and A. on X to be
compatible if ~.i_:r:. d(SAx,,, ASx,,) = 0 \vhenever ( x,, " is such that ~i_:r; Ax" =
Sx,c = t some t;::: X. It is easy to see that if self-maps S and A. on X are r:-+:r:
compatible_ then A.Sx=SA.x wheneer x EX is such iLr = Sx. Self-maps which
commute at 1ueuu:: points are knmvn as weakly compatible [3]. In an attempt to
weakly compatible, Renu
x (x = 2 or x > 5) A't = .
6 (2 < x ~ 5)
· the statement that every compatible pair is
Sar:jay Kumar [ considered self-maps:
Sx=
x (x = 2) 12 (2 < x s 5)for all x EX,
-3(x > 5). (1)
\vhere X= [2,201 with usual metric d(x,y) = x - yj for all x, y EX. It was claimed
that the mappings are not compatible.
The follov.-ing lines reveals that their claim is not true. The maps A and S are, in fact, compatible. For,
22
O (x = 2)
d(Sx,Ax) = ~ 6 (2 < x s 5) and d(SAx,ASx) =
3 (x > 5i
(x=2orx>8)
< x s 5) : 9- x (5 < x s 8).
so that d(Sx11,Ax,,) ~ 0 as n----* x whenever \ x, v·· h' ')~ c A is sue tnat x., = ~ tor
all n and hence d(SAxn ,ASxn )-+ 0 as n-+ :r..
. 2 (x = 2 or x > 5 i . . . . However, if we redefine A as A;: = < ; · a routme computatwn
6 (2 < x s ;'.). -
reveals that A. and S are not compatible. In this case. x= 2 is coincidence
point for A and S at which they commute and hence tA.,S is ·weakly compatible.
REFERENCES i l; Renu Ch ugh and Sanjay Kumar. Common Fixed Poin:s :l::1r Compatible Maps. Proc. Indian
Awd. Sci. !Math. Sc11. 111(2) 1:20iJ1 :241-:247. ':21 Gerald Jungck, Compatible ~lapping and Common Fixed Poims. f>-;t_ J. Maih. tuul. Math. Sci.
9(4} 11986). 771-719. [3] Gerald Jungcck and B.E. Rhoades, Fixed poim for ::et valued functions without conunm
J. Pure Appl. Math. 29 (3) (1998), 227-238. Indian
Jnanabha. '2007 Honour Professor S.P Singh on his 701
h Birthday)
A NOTE ON PAIR\VISE SLIGHTLY SEMI-CONTINUOUS FlJNCTIONS By
M.C.Sharma and V.K.Solanki Department of :\fathematics
XR.E.C. Collage Khurja-203131, Uttar Pradesh, India
U?.ecei0ed : .'\larch 15, 2007)
ABSTRACT In this paper we introduce concept of pairwise slightly semi-cintinuous
function in bitopological spaces and discuss some of the basic properties of them.
Several eexamples are provided of illustrate behaviour of these new classes of
functions.
2000 Mathematics Subject Classification : 54E55. Ke~-words and Phrases : i,j .1 dopen set, pairwise slightly continuous, pairwise
slightly
regular.
pair,•;ise almost semi-continuous, pairwise semi()
·v,·eakly semi-continuous, pairwise s- closed, pairwise ultra
L Introduction. Kelly[5] initiated the systematic study of bitopological
spaces. A set equiped with two topologies is called bitopological space. Continuity
play an important role in topological and bi topological spaces. In 1980, Jain ( 4]
introduced the concept of slightly continuity in topological spaces. Rece11tly Naur
defined a slightly semi-continuous functions as a generalization of slightly
continuous using semi-open sets and investigated its properties. In 2000,
a note on slightly semi-continuous functins in
topological spaces.
The object of the paper is to introduce a new class of functions
called pairwise slightly semi-continuous functions. This class contains the class
of pa.irv.rise continuous functions a.11d that of pairwise semi continuous functions.
Relations between this class and other class of pairwise continuous functions are
obtained.
Throughout the present paper the spcaes X and Y always represent
bitopological spaces lX,P1,P2 .' and n:Q1,Q2) on which no seperation axioms are
assumed. Let S = X. Then S is said to be (i, jJ semi-open [8J if Sc P1 -Cl (P1-
InttSJJ twhere P1-Cl(S) denotes the closure operator with respect to topology Pi
and Pi-Int( SJ denotes the interior operator with respect to topology P1, (i, j= 1,2,
i= j) and its complement is called (i, j) semi-closed. The intersection of all (i, j)
24
semi-closed sets containing S is called the (i, j) semi-closure of S and it v.'ill be denoted by (i,j) s Cl(S). A subset Sis said to be (i,j) semi-regular if Sis both
jl semi-open and (i, j) semi closed. A subset Sis said to be (i, j) semi 8-open if S is the union of (i,j) semi-regular sets and the complement of a j i semi 8-open set
is called (i,j) semi e-closed. A subset Sis said to be (i, j) clopen if Sis Pi-open
and Pi-closed set in X.
Ir: this note we denote the family of all li .. jl semi-open \resp. P1-open, 1.i, j;
semi-regular and (i,j) clopen of (X.P1,P2) by (i.j1SOrXHresp. P;-open(Xi,li,j SRlX and li,j)CO(XJ), and denote the family of (i,jJ semi-open 1.resp. Pi-open, ti,j! semi
regular and (i,j) clopen) set of 1X,P1,P2 l containing x :i,jiSOCX,x J resp. Pi <X,x i,
(i,j)SR\X,x) and ri,jJCOtX,x). i,j=l,2. i=:=f.
2. Preliminaries. Definition 2.1. A function (L'K_,P 1,P2)-+ Y:Q 1,Q2 1 is said to be pairwise-semi
continuous [8] (p.s.c.) (resp. pairwise almost semi-continuous I 1p.a.s.C)[12], pain.vise
semi 8-continuous (p,s,8,C) [12] and pairwise weakly semi-continuous
(12]), if for each xc: X and for each VE Q1CvJ(x)) there exists U E (i,j)SO{X,x
such thar f(U) c V(resp. f(U) c Q, -int(Q) -Cl(V)) f(i,j
f(U)c QJ -Cl(V)).
Definition 2.2. A function {"(X,P1,P2
continuous \p.a.C.) [2] (resp. pairwise
continuous (p.w.C.)[2] if for eachxc:X
such that f(U) c Q1.-Int\Q,-C[(F1 l :resp.
I -
Definition 2.3. lll] A function r1)(P
continuous (p.sl.s.C.)(resp. pairwise slightly continuous
)cQJ-
and for each VE (i,j)COCY, {(;:)},there exists U E 'i,j)50(X,xi (resp. Uc: P/)(x)) such
that ft U -:: V i, j = 1,2 and i¢j.
A function hX,P 1,P 2)-+(Y,Q 1,Q2) is said to be pairwise slightly semi-
continuous (resp. pairwise slightly continuous) if inverse image each J-dopen
set of Y is (i, j) semi-open (resp. Pi-open) in Xi, j = 1,2, 1:=.f. The following diagram is obtained in (11] :
p. C => p.a.C => p.9.C. => p.u.:.C __, p.sl.C ii v
p.s.C =>
JJ
p.a.s.C => p.8.s.C => diagram 1
·' v -\..~
p.w.s.C =>
Remark 2.4. It was point out in [11] that pairwise slightly implies pairwise ~lighltly semi-continuity, but not conversely. Its counter examole are not
1c; L. ,
g1t:en u:r
Example 2.5. Let X={a, b, c . P 1={o,X, {a},{b},{a,b}}, P 2={o,X, {aUa,bH and
Q wr l'' Q ' "T " 1 'T'h th . f (X p p '1 i Y.Q Q ) . L= .·1.. a 1 !, - 2 =10, L ':l'J,c ;, _en emapp1ng :, , 1, 2 ,-7,, 1, 2 , is
pairwise slihtly semi-continuous but not painvise slightly continuous for t 1({a})
is semi-open and (i, semi-closed, but not P1, -closed in (X,P,,P.7 ).
' -Theorem 2.6. The pairu;ise set-connectedness and the pairwise slightly continuity are equivalent for a surjectir.:e function.
Proof. A surjection f(X.P:.P2\-7fYQ 1.Q2 l is pairwise set-connected if and only if
is i, clopen in X each i, f' clopen set F of Y. It is easy to prove that a
function X,P:,P2 -7n:Ql,Q2 : is pairwise slightly continuous if and only if t 1(F)
is P 1-open in X for each Ii, j! clopen set F of Y. Therefore, the proof is obvious.
Theorem 2.7. For a function f:(X,P 1,P2J-+(Y,Q 1,Q2) the following are equivalent:
f is pairv.ise slightly semi-continuous, r- 1r1/ E . j;SQiXJ for each VE(i,j)CQ(YJ,
kl r- 11/ is i, semi-open and li,j) semi-closed for each VE(i,jJCO(Yl.
3. Properties of pah"'\r.ise slightly semi-continuity. Theorem 3.L The follov.·in;z are equiYalent for a function f(X,P 1,P2 )-7(Y,Q 1,Q2l:
;,,,aJ semi-continuous,
For each xE:X and for each (V)E(i,j)CO(Y,f(x)), there exists UE(i,j)
SR(X,x) such that f{Ul = \: (c) For each xEX and for each (V)E(i, j)CQ(Y, f(x)), there is UE(i, j)
SO(X,x) such that fii, j )sCl :r1; = V Proof. (a)=> (b). Let XEX and v E(i,j)COiY, /hll. By theorem 2.7, \Ve have r- 1(V)E
\ij)SR(X,x). Put U=f- 1CV:J, then xEU and f([J) = V
(b) => (c). It is ob\ious and is thus omitted..
(c)=>(a). ff
Definition 3.2. A
. then U,J ~ .i.j)SO{X) .
sDcce · X,P 1,P 2 is called Pairwise semi-T2 [6J resp. ultra Hausdorff or pairwise UT2) if
for each pair of distinct points x, y of X, there exists a P 1-semi-open (resp. P 1-
clopen) set U and a F 7-semi-onen resp. P.7-dopen} set V such that xE:U, yEVand - . -(}"'Vr=O.
Pairwise s-normal[7] {resp.pairwise ultra normal) if for every Pi-closed
set A and P1-closed set B such that AnB=ijl, there exist UESO(X,P) (resp Co(X,Pi)
and VESO(X,P)(resp.(X,P)such that Ac U, B c Vand UnV=cp,where i,j=l,2, i;z::j
Pairwise s-closed (resp.painvise mildly compact) if every (iJJ. semi regular (resp.(i,j) clopen) cover of (X,P 1,P2) has a finite subcover. i,j=l,2 and i=t=j.
Theorem 3.3. If f: rX,P1,P2)-7rY,Q 1.Q2J is a pairwise slightly semi-continuous
injection and Y is painvise UT2 ,then Xis pairwise semi-T2.
26
Proof. Let x 1,x2 EX and x l't:x2. Then since f is injective and Y is pairwise UT 2.
*{(x') land there exist, \l,,V'?E (i,jiCOi Yi such that th, i E \!,, fh.; 'E \/.1 and \l1 -V.; =o. -- l ... l ..l. - - - .;;..
By Theorem 2.7, xiEt1\Vi)Eli,jlS01X1 for i=l,2 and f 1 1'l~ 1 \l2i=o. Thus Xis
pairwise semi -T 2.
Theorem 3.4. If f"(X,P 1,P2J-H'r:Q 1,Q2 is a pairwise slightly semi-continuous.
Pz-closed injection and Y is painvise ultra normal, then Xis pairwise s-normal.
Proof. Let F 1 and F 2 be disjoint \P 1,P 2 .1-closed subsets of X. Since {is P z-closed
and injective, f(F 1) and f(F 2; are disjoint { Q 1,Q 2 )-closed subsets of Y Since Y is
pairwisE ultra normal, f(F 1) and t1F2 l are separeted by disjoint Pi-clopen sets \11
and Prclopen \l2, respectively. Hence Fict 1CV), f 1CViJE · j)SO(X) for i=l,2 from
Theorem 2. 7 and f 1 ( V 1)nt1 CV2 l = o. Thus X is pairv.ise s-norrnal. Theorem 3.5. If {-lX,P1,P2 l-7(Y,Q 1,Q2 l is a pair.vise slightly semi-continuous
surjection and (X,P 1,P 2 ) is painvise s-closed, then Y is pair;i,ise mildly compact.
Proof. Let {Va IV a: E lij )COCfl,aE \ be a cover of''( Since (is painvise slightly
semi-continuous, the Theorem 2.7 {f·1Vu' o:E \ is a }! semi-regular cover
of X and so there is a finite subset \ such that
X=LJr 1(V,,) uc\ 0
Therefore,
Y= LJV" uE\.u
(since f is smjective
Thus Y is pairwise mildly compact.
Theorem 3.6 If f(X,P1,P2 )-7('/Q1,Q2) is painvise semi-continuous and
Y is pairwise UT 2, then the graph G(f) of (is i, Ji semi 8-closed in the bitopological
product space Xx Y Proof. Let (x, y) e: G({J, then ;i:;:: f(x). Since Y is pairwise UT 2_ there exists VE u, J
COCY, y) and WE(i, j) COCY, f(x)) such that V'lW=o. Since l is pairwise shghdy
semi-cor cinuous: by the Theorem 2. 7 there exists U E · j; SR x such that
f(U) c W. Therefore f(U)(f\l = o and hence
U E (i,j)SR(X,x) and VE(i,j) C01Y: y),
,...., = o. Since
(x,yl EUx Vand Ux VE (ij)SRCX x Y). Hence Gl{l is ii.Ji semi El- closed.
Theorem 3.7. If f: (X, P 1, P2 )~(Y, Q 1, Q2 .l is pairwise slightly semi-continuous
and (Y, Q1, Q2J is pairwise UT2, then A={(x1,x2 ) /f(x1 =ftx2 ! is i_.j)semi -El
closed in the bitopological product space X x X.
Proof. Let (x1,x2 ) ii A Then f(x 1 h=f(x2 ). Since Y is pairwise UT 1 , there exists V 1
E(i,j) COO': f(x, )) and Y9 E (i,j) CO(Y, f(x 9)l such that V1 '""'\l)=o. Since fis painvise l - - . -
27
semi-continuous, there exist U 1,U2 E Ci,j)SR(X) such that xi EU; and f(Ui)c
~-for i=L2. Therefore, 1x,.x2 EU~ u:.!:. U1:<UzE(i, j) SR(XxX), and iUlxUz)n
A=o. So A is i, j .l semi 8-closed in bi topological product space XxX
Definition 3.8. [3] A bitopologicahX, P 1, P 2 ) is said to be pairwise extermally
disconnected if P2 -closure of each P 1-open set of (X, Pl' P 2J is P 1-open.
Lemma 3.9. Let (X, P 1, P 2 ) be pairwise exteremally disconnected space, then UE
i.j)SRCXl if and only if L-E j)CO!K1: i,j=l,2 and i :;t:j.
Proof. Let lI::= . j1SRLX::. Since Uc 1iJ1SO(X1,PrCl iU)=PrCUPi-lnt (Ul and so
X'. Since [J is ; i, j 1 semi-closed, P 1-Int ( UJ = U = P rCl ( Ul and hence
The convers is obvious.
Theorem 3.10. If f: (X, P 1, P 2 l--,1-C'i: Q 1, Q2 ) is pairwise slightly semi-continuous,
and X is painvise exteremally disconnected then f is pairwise slightlY. continuous.
Proof. Let xEX and VE · j!CO(Y,f(x)). Since fis pairwise slightly semi-continuous
Theorem 2.7. there exists U E (i,j)SR(X,x) such that f(U) cV, since Xis pairwise
exteremally disconnected by the Lemma 3.9,U E (i,j) CO (X) and hence (is pairwise
slightly continuous. Difinition 3.11. )-~ f : 'X, P 1• P 2 )--,1-(}.', Q 1' Q 2 ) is called pairwise almost
ti-::;;en11 continuous (p.a-st.8.s.C. (resp. pairwise strongly 8-semi continuous
\p.st.0.s.C.) if for each x EX and for each VE Q/Y,f(x)J, there exists Uc ti,jlSO\X,xl
such that f((i,j! sCl!U)) = i,j)sCl(1/) (resp. f((i,jlsCl(U))cV
Theorem 3.12. If f: {X, PI, P 2J--,1-(Y, QI, Q2 l is pairwise slightly semi-continuous
and (Y, Ql, Qzl is pairwise extermally disconnected, then r is pairwise almost strogly
e-semi-continuous.
Proof. Let xcX and VE Q/Y,frx.iJ, then · ·1sCl1 = Q;-lnt{Q1-Cli\ is (i,j) regular
open in r Y Q:. . Since Y is painvise ex:teremally disconnected. i, j l sCU Vl E lij l
CO (Y). Since (]s semi-continuous, by Theorem 3.1, there exists
u E li, J so U) = . j)sCl{ VJ). So r is pairwise almost stroghly 8-semi-continuous.
Corollary 3.13. [11] If f:(X,P 1,P2 J-7C( Q 1,Q2) is pairwise slightly semi-continuous
and Y, Q1, Q2 ) is pairwise extermally disconnected, then (is pairwise weakly semi
continuous.
Difinition 3.14. A bitopological space !X, P 1, P.2 l is called pairwise ultra regular
if for each UEPi(XJ and for each XE U, there exists OE U,j) CO lXJ such that xEOcU
Theorem 3.15. If {:(X,P1,P2)-7(Y,Q 1,Q 2) is pairwise slightly semi-continuous
and a:Qi,Q2) is pairwise ultra regular, then f is pairwise stroghly 8-semicontinuous.
Proof. Let xEX and V EQ/Y,f(Xll. Since rY, Q 1, Q2 J is pairwise ultra regular, there
is WE i, jJCO Yl such that ({x)EWc:::V. Since r is pairwise slightly semi-
28
continuous, by the Theorem 3.1 there is Uc (i, j) SO (X, x J such that f{(i, j)sCl(
cW and so f(i, j)sCZUJ)) cV. Thus f is strongly 8-semi-continuous.
We have the following diagram:
fl]
p.5.C
J P.C ii v
=
P.s.C =>
D
P.st. e.s.C
P.a.C = v~
P.a.s.C => -'·
=> P.st. B.s. C
P.G:C=> P.w.C => -u v
P.s. G.C => P.u:.s.C =>
REFERENCES S.Bose, Almost open sets, almost closed, 8-conrinuous 3..:1.d alffio:;t bitopo!ogical spaces, Bull, Cal. Math.Soc; 73 '. 1981 '.345-354.
P. sl.C
v
Psl.s.C
compact mapping in
[ 2] S.Bose and D. Si:nha, Painvi.se almost continuous map and weakly continuous map in bitopological
spaces, Bull, Cal. Math. Soc, 74i1982;, 195-206.
[3] M.C.Datta, J.Austr. lvfath. Soc.: 14 \1972), 119-128.
[ 4] RC.Jain, The Role of Regularly Open Sets in General Topology, Ph.D. Thisis, :'.l.feerut
Institute of Advanced Studies, I\foerut-Indiail980J.
[5] J.C. Kelly, Bi.topological spaces, Proc. London Zvfath. Soc., 13 1961),71-89.
16] S.N. Maheshwari and R. Prasad, Some new separation a.xi.oms in br!opuli{cg±c3.l ~s. .li1Iilillten:i84'.u:c
12 (27) ( 1975), 159-162.
!7!
[8]
[9]
S.N. Maheshwari and R. Prasad, On pairwise s-ncnrsl s2p:es
40.
"L"'.,~f;s:':!P. _'tfaif: .. J~~ 15 t l9;5L 31'7-
S.N. Maheshwari and R. Prasad, On pairv;ise irresolute fon.:-:im1s. _lfo:the?natica. 18
172.
169-
T.Noiri and G.I.Chae. A note on slightly serni-con::im.ious (2)(2000), 87-92.
BuU. Cal. },foth. Soc_; 92
[10) T.M.Nour,Slightly semi-continuous function, Bull. Cal. lvfath. Soc; 87 (1995), 187-190.
[ 111 M.C'. Sharma and VK.Solanki, Pairwise slightly semi-continuous function in bitopclogi.c:al spaces. [communicated]
( 12] VK.Solanki, Almost sewi-continuous mapping in bi to pc logical spaces. The _lf athemniics Education XLII, No. 3 (To appear)
29
Jria;,11ahha, 37, 2007 to Honour Professor S.P Singh on his 701
" Birthday)
N-POLICY FOR MJG!l lVL.c\CHL.'i"'E REPAIR PROBLEM WITH STAl'IDBY Al'ID COM:M:ON CAUSE FAILURE
By
M.Jain
Department of Mathematics, LI.T. Roorkee, Uttar Pradesh, India E-mail : madhufma(i"1 iitr.ernet.in, madhujain~l,sancharnet.in
and Satya Prakash Gautam
Department of Mathematics, Institute of Basic Science Dr. B.R. Amedkar University, Agra, 282002, Uttar Pradesh, India
E-mail : gautamibs({~ gmail.com meceiued : April 5, 2007 J
ABSTRACT In this paper, 'we investigate an optimal N-policy for a single repairman
machine Poisson arrivals and general service time distribution. We analyze the machining system consisting of M operating machines along \vith S cold standby machines. For the normal operation, lvf machines are required; however, the system can also function with at least in m( <lvf) machines in degraded mode. The machines may fail indi\idually or due to comman cause. The repair times of the failed machines are independent and identically distributed random variables with general distribution. The repair rendered by the repairman is assumed to be imperfect. The governing equations are constructed by introducing the supplementary variable corresponding to remaining repair time. To obtain the steady state recuz·si;,-e method is employed. A cost function is developed to at minimum cost. 2000 Mathematics Subject Classification: Primary 90B50; Secondary 90B90. Keywords : Queue, iV standby, Recursive method, Supplementary variable technique, Common cause failure, Cost function.
1. Introduction. The modeling of machine repair problems has found increasing attention in recent years due to their. practical applications in several areas, such as in manufacturing systems, computer systems, communication networks, etc .. For machining systems, the provosion of spare part support is common to improve the efficiency of the system but it increases the cost of the system. In many realistic situations, the server may start the service after accumulation of the pre-assigned number of the jobs. This policy seems to be cost effective. In the peresent paper, \Ve study an optimal N-policy for M!G!l machine repair problem with spares under the steady-state conditions.
In a machine repair problem, if at any time a unit fails, it is sent for repair to the repair facility. The repairman can repair only one failed unit at a time. If an
30
operating unit fails, it is immediately replaced by a spare unit if available. In many practical situations, the spare units are needed to operate the system continuously over a long time period. These situations are applicable in different areas such as production lines, airlines, manufacturing organizations, tele communication, etc ..
Machine repair problems have been studied by several research worders in different frame-works. Some of them are Cherian et al. (1987),jain et al. (2001), and many more. The policy, in which the repairman will not initiate the repair of failed machines till N failed machines are accumulated to abate idle time repairman, is termed as N-policy. Several attempts have been made to suggest control ;1olicies in this field from time to time.Choudhury ( 1997) developed a queueing system with general setup time and poisson inputes facilitated under IVpolicy. The queueing system with finite source ai"ld '•Varm spares under was investigated by Gupta (1999).
Recently, N-policy for redundant repairable system with additional repairman was investigated by jain (2003}. Wang et al. (2005) presented the maximum entropy analysis to an lvf!G 1 queueing system \vith unreliable server and general startup times. Kim and Moon (2006) discussed an MiG 1 queueing system under a certain service policy. A two phase batch arrival retrial queueing system with Bernaulli vacation schedule was explored by Choudhury (2007).
Numerous queue theorists developed machine repair problem having individual as well as common cause failures. Hughes{1987) i~'J.le
of common cause failure in machining problem. jain et al. (.2002) investig&ted a flexible manufacturing system Dhillon and Li (2005) suggested stochastic analysis systems v:ith com.mon cause failure and human errors. Jain and ~fishra(2006) studied multi-stage degraded machining system with common cause shock failure and state dependem rates. Ehsani et al. (2008) proposed a mod.el for reliability evaluation of deregulated electric power systems v1ith common cause failure for planning applications.
The purpose of our study is this paper is to provide a recursive method to determie the steady state probability distribution of the number of failed units in the M/G/l system operating under IV-policy. In section 2 ;,ve outline some assumptions and notations in order to construct the mathematical model of the system. · ~ction 3 covers the analytical expressions if all probabilities for different states. In section 4 some important performance characteristics such as average number of failed machines in the system, the probability of repairman being idle, machine availability etc. are establised. In section 5 we determine the average length of the busy period, average length of the cyclP :ind average length of the idle period. Also we construct the cost function in term,.. total expected cost per unit time. Finally conclusion is drawn in section 6.
2. Model Description. We consider general service system consisting of KuVI operating + S cold standby) machines under the supervision of single
31
repairman to maintain efficient operation for long run. The model is developed by considering the follovving assumptions: 11 1 life time of operating machines follows exponential distribution with
mean li .. The system may fail at any moment due to common cause with exponential failure rate i.e. o:(O s a s l } is the probability of recovering the failed machine. The renewed machines become as good as ne\v ones. The system >'Vill be in working state until there are m operating machines. If an operating machine fails, a cold standby machine replaces it with negligible S\vitchover time pro·,ided it is a aYailable at that moment.
Iv) The repair times of the failed machines are independent and identically distributed li,i,d) random variables having general distribution function. The repairman follows N-failed machines in the system and continues the repair until he repairs all the failed machines.
For modeling purpose, \Ve use the following notations: N Threshold level B K B(u)
uw bl
\: ld
BJ
Random variable denoting service time Total number of machines, K=M+S.
l of B Probability density function (p.d.f) of B Remaining repair time for the machine at time t Mean repair time Probability of recovering of the failed machines Laplace- Stieltijes transform (LST) of B
order derivative of 8*(8) with respect to 8 Common cause failure rate. Degraded failure rate, when n~ S + L
Let QU} and L be the random variables denoting the status of the. repairman at any instam t and the number of failed machines in the system respectivley. Define
(t) = 0, if the repairman is in idle state at time~ Q · · 1, if the repaiman is no \vorking state at time t ·
Using these notations, we define the following transient probabilities: P0_0{t) =Prob {Q(t)=O, L(t)=O} P0.IJ(t) =Prob { Q{t)=O, L(t)=O, 0Sn<N} Pi,n(t) =Prob {Q(t)=l, L(t)=n, lsnsK-m}.
For the analysis purpose, we follow the supplementary variable technique [cf. Cox, ( 1995)]. By introducing random variable U corresponding to remaining repair times (u) for the failed machines in repair, we shall construct the equations of different states of the system. Let us denote
32
P 1 n(u,t)du =Prob {Q(t)=l, L(t)=n, u<U(t)s,u+du, u2:0, l5on5,/{-m}
P1, 11 (t) = J: P1.11 (u,t }iu.
3. The Analysis. Considering the state of the system at time t, we obtain the tansient state equations governing the model as follows :
!!:_ P.1, (l (t ) = -(MA + ),,_}P.fJ 0 ( t ) + P, r ( 0 ,t) dt . - . ' . ··'
!!:_ P. ( 1 - -( M ' + ' )P dt O.u t' - - /, · /,c 0,12 + lvVP0_11
_ 1 (t), 1 s, n :-::; N -1
a a ' . 3t - au )P1.1 (u,t) =-(MA.+ 2c)Pu (u,t )+ Pu(O,t ~(u)
c 3\1 ( ) ( . { . - - - P1 ,, u,t = - lvD. + /,_, )P n u,t)+ al1,f;.P, ,,_. Ot au j ., ' "·. ·-· .
2:-::;n:-::;N-1
+ PL•-l
la_ a -1 Ot au _;Puv (u,t) =-(Ml,+ )P1.s t) ..... wV.[l.Pi.s-i (u,t )-'--Pu--i
+·Afl.Po.N-1 (u)
' u D ·1 ·
l---JP111 (u,t)=-(M!,+l.,)P,,. at au) ' '·· a.li,fi.P,. __ . i u. t
N+l:s;n:s;S
(~-a I Ot Ou )P1,n (u,t) = -((K -n)'Ad + /,JPiJ, a(K-n+
+ PLn+l (O,t )b(u ), S+l:s;ns,K-m-1
:t Pl,K-rr 1t) = -(rnt.d + \. )Pl,K-m (t )+ o:(m + 1)1,dPl.K-m-:
:t PF(t)= o:(K -m)fi.dPl,K-m(t)+ ~ {(1-a)M/, + !.JP1,i
K-m-1 .V-1
+ L {(1-o:XK -i)t,J +I., }P1)t )+ L\P0)t) i=S-1 i=O
For steady state probabilities, we deonote
PF =~~~PF (t}, Po,n = ~~~ Po,n (t}, 1 :-::; n 5, N -1
... ( 1
... 141
... (6J
... (7)
... (8)
:-'/"'::: P1 n \u) = limPi." (u,q; 1::; n::; K -m -··. ' :____,.:--
P __ =hm
-[ 1,.~4 :=. _1b(u) .
From equations i1H9>. the steady state equations are obtained as
-1.M/~ ~ - -P,.10)=0
- .\1 ;_pJ n-1 (0) = 0. lsns;N-1
d P . -- t1'1·· - . -)P ' ) P --d u<uJ=-vl"./.-/., :_;.t,u.~ i.2 u
d --Pi_,
du
- !!:._ PLS ) = du
d
d
du
=
""/.c )P1 Ju )"-- wW/PLn-i (u) + Pl.n-l (0 }b(u), 2 s n s N -1
· - /;., )P1 s (u )+ m\f )Pi.s-i (u )+ M /Po.N-i (u )+ Pi.N~i Ob(u)
_;_ 7c.L"'vll..P,_,,_ 1 (u)+P1J1
-;. 1 (o)b(u),N +1 s n s S
--:- !~.~ )-'- cr.(K - n -'-1 )1.dPLn-l (u) + Pi.,,_,, 1 (0 )b(u),
S.J._lsnsK-m-1
+ / .. c )PLK-m -"- o:(m + l)/,dPLK-m-1 = 0
s a(K-m)!.dPlJ(-n;-'-)
i=l
;_'-.;:_1
K-1n-l - 0 ·)~11· -'-!. 1·P --'- '"'\l(l-a\JK- 1
v. . .:v • • ·c I l.i ' .L ! II, • ~.::;:..'):-l
33
... (10)
. . .llll
... (12)
... (13)
... (14)
... (15)
16)
(17)
+ ) i.cPo,: = 0 ... (18) i=O
Equation ( lOJ yields
Pu (0) =(Mi.+ !.J.P,)_o . . .. (19)
Putting n= l, equation (11) prn;,ides
p - },;fi_ p O.l - Jvfi. + • OJi • ... (20)
Now putting n=2, equation ( 11) yields
:2
1Vfi. I Po_o Po.2 = lvf),.,,. , ... (21)
Similarly by using rcursive approach, we obtain
34
~l.n = MJ..
Po.o l::;n::;N-1 Ml-+!-..,
Further let us define Laplace-Stieltjes transform as
B*(8)= [e-e"dB(u)= r e-e"b(u)du .
P1~,,(e)= r e-Hupl,n(u)du,
so that, we have
l<n::;;K-m
.. Jf ()d Pi.11 =P{:n(O)= 0
P1,11 u .u, L;n::;;N-m,
,e-Bu :U P1,12 (u)du = 8P1~n(e)-P1,n(O),
Puv-i (u) = Po,N-I b(u) .
... (22)
. .. (23)
... (24)
... (25)
... (26)
. .. (27)
Taking Laplace-Stieltjes tansforms (LST) on both sides of equations (12) to (16), we have
(Ml.+ 1-..c - 8 )P1: 1 (e) = Pl.2(0 )B ':' (e )- Pu (0)
(MA.+ A.c -8 )P1~n (e) = a.M/.P1: 12_ 1 (e)+ Pi,n+l (O)B *(B)-PL>r (UJ, 2::; n::; N -1
(MA.+ A, -e)P1:N(8) = a.M:U{v-1 ~'llrft .PO ..... _ 1 B * P,_v-: .:::!;;
-Pr r +l::;n::;S (MJ.1. + l.c -8 )P1:,, (8) = a.Ml.P1:n-1 (tl)+ f>r .. r:-:
[(K-n)l .. d +l.c -e]l{J8)=a(K-n+ )- P1M1 (O)B * -PL,,
S+l::;;n::;K-m-lFrom equations (19) and (22), we get
/
n f 1 \11
M' \11
r0
,, =! A . \Mf..+J..,), Ml.+!. l Pu(O), l::;n::;N-1 \_ c /
Using equation (33) in equation (30) and adding equations (28)-(32), we get
S K-m-l ·
{MA.+A.c e)IP1:,,(e)+ I[(K +n)! .. d +Ac -e}Pl~n(e) n=l n=S+l
S K-m-1
=a..MJ1.IP1~,,_1 (8)+aA.d I(K-n+l)P1~n-i(8) n=2 n=S+l
I( ,Nl
K-m-l MA. + I P1,11 + 1 (o)B*(e)+ j 1.1Pu(0)-1 .
11=1 • MA.+ Ac / I - -
... (31}
... (32)
... (33)
... (.34)
35
Now using equation (33) in equation (30) and setting e =Ml.+ I., in equations
(28H3U, we obtain,
P, _ = , .. -4iPu(O)-a.A1iJ{,_1(lYfi.-) <.n-i B"'(M" · · . . < r.-1.c} l::::n::::S ... (35)
where
l\1i. n=N M'"t.+ b · d P ( 1 0 f ot .envise an 1 " : 1 = or n < 1. 0, ., '
- Now substituting e = (K - n )1.d + l.c in equation (32) we get,
PLn-l
PL,,{0)-a[(K -n+l}i.dJPl:n-l[(K -n)l.d +i,.J
= B * [(K - n }1.d + !.J S+l:::;n::::K-m-1- ... (36)
Differentiating equations 128)-1'.31) j times w.r.t. '8' and setting 8 =Mic+ f,_c we have
.~l ~ - = - 1 lrP_ -~OlB ,;{,; ~ ..... J~."\"'l \ -
J . ) ·p (O)B -·-UI (M· . ) +Ac + lp l,l ··-. A+ Ac +
:11· n~u1 (M· • )1 OU:v l..rLn-1 _1 A~ '·c • ' 2 ::; n ::::: S - 1, 1 ::::: n ::::: S - n - 1 ... (37)
. dj & B *IJ! (8) = deJ B * (e) is the ;th derivative of B '-' (e). where P":?l(e)· = p"" L,.. L.
Again setting e = -r)! .. +).0
, in equations {28)-(31), we get
Pl~n[(kf - + - r -'- + tfif; 1 (O)B * {(lv.f - r)I" + !,_J
+ P1~n+1 (0 )B" +1 .. '.-P:..r. l:; n:; S, 1:; r::: M - m - l. ... (38)
Again setting e = (M - r )I.a + , equation (32) yields
P1~" [(kf - r )?.d + )\."] = 1,.., l )• [a(K - n + l)?.dPl~n-l {(M - r )A.d + "-c} + Pl,n+l (0 )B * -n..,-r_1.d .
- r )i_ d + i. ,- }- Pi.n (0 )] , S + 1::; n::; S + r -1, 2::; r::; M - m -1. ... (39)
Now Pi.2 (0hPLS(O), ... ,Pu-m (0), can be obtained recursively with the help of equations . .
(35)-(39) in terms of P 00.
Again setting e = 0 in equatfons (28)-(32), we have
36
p1
'
11
(0) = cu.VJ/,Pl~n-1 ( 0) + PLn~l (0 )- PLn (0) Mio-I·. c
where B '(0)= 1.
Also
" ( aMAP." (0) , p Pl,n 0)= Ln-1 ' Ln i\V}~
iVli. +
and
P1,11\0)= rJ.(K-n+l)i.c1Pi.,, i(O)+P1n-1 (K -n)l.d--'-
l::;n::;X-1
X::;n:S;S
, S-l::;n:S;K-m-1 ... (42i
Now PLn (0),(1:::; n :S; K -m), ca.n be determined respectively using equations (40)
(42) in twms of P0_0.
From equation (4,17), we obtain
a.(m ~ l)i.dPi.K :. pl.K-m = (mi.d + )«·)
The value of P O,O can be determined by using normalizing condition as
N-1 K-m
_LPo,n + _LP1~n(O)= 1 . J: ,·~
11=0 n=l
The failure probability of the system can be
N-l K-m
Pr= 1- }_::P0,11 - _LP1C.Jo) <O n=l
4. Some Performance Indices. In this section, we present some performance indices so that the system designer may ensure efficiency and effectiveness of the system for future design and development. Now, we formulate the performance indices in terms of queue size distribution obtained in previous section. Let us denote the performance metrics as follows : E(N) Average number of failed machines in the system. E(Nq) Average number faild of machines that are waiting for repair in the queue .. E(O) Average number of operating machines in the system. E(S) Average number of standby machines in the system. P(l) 'I he prob.ability of repaiman being idle. P(B) The probability of repaiman being busy. MA Machine availability. The above performance indices can be given in terms of queue size as follows :
N-l K-m
(i) E(N)= _LnPi)-'1 + _LnP1,1 ... (46) 11=0 n=l
37
S-1 K-m
J= I(n-l)Po.n + L(n-l)P1 " ... (47) n=i !<. = ~
K-m = l¥I - L (n - S)Pi.n ... (48)
n=S+l
s E(S)= }(S-n)P ..
., .__.. . l.,h . .. (49)
.\- - 1
VP = I?r: n = NPoo ... (50)
(vi) P(BI= 1-P(I) ... (51)
1 _\--1, , K-m 1 AtA = I{K -m -n)P0_11 + I(K -m-n)Pl.,, i.
- m. ,._:-; n=l J . .. (52)
5. Cost Analysis. We assume that E{B), E(C) and E(l) represent the average length of the bru,,J'' period, average length of the cycle and average length of the idle period, respectively. Then average length of idle period is obtained as
E(t)= M;• .. . .. (53)
The long run fraction of time for which repairman remains idle and busy, are given by
and
E(I) s E(C) = InPo.o.
n=1
E(B) =1-Y E(Cl .'.;::"i
. .. (54).
... (55)
To determine the optimal value the system, we construct the cost function denoting the total expected cost per unit time as given below:
Tc( J>.r S) C E(B) , C E(I) . C ( ~· · .rv. = 0-(-)' r-(-)-r ,E.:.\~-"-, . . ·EC EC ... , +C5 ) E~) +(Ml.+!,JP1~K-m(O) ... (56)
where different cost elements used are as follows: C
0 = Cost per unit time for keeping the repairman on
C,- = Cost per unit time for keeping the repairman off. I
Cn. = Holding cost per unit time per machine present in the system. Cb = Break-down cost per unit time for turning the repairman off. C
5 =Start-up cost per unit time for turning the repairman on. 6. ·Conclusion. In the present paper we have obtained the steady state
solution for the N-policy M!Gil machine repair problem with cold spares. The
38
concept of the common cause failure along \vi th individual failures is incorporated, which is common in many real time machining systems. The suggested recursive method using supplementary variable technique to determine the steady-state probabilities can be easily implementable to quantify various performance indices as well as cost function. Our study provides good tradeoff for gaining initial insight of system's performance and cost incurred so as to determine the optimal senice parameters. The concerned model is more sophisticated and versatile in our real life situations as it includes common cause failure and imperfect repair performed by a single repairman.
REFERENCE l ll J., Cherian, M. Jain and G.C. Sharma, Reliability of a standby system with repair, Indian J. Pure
Appl. Math., 18 No. 12, \1987l 1061-1068. [2] G. Choudhury, A poisson queue under :V po!icy wi:h a general semp time, Indian J. Pure Appl.
Math., 28 t 1997), 1595-1608. [3] G. Choudhury, A hvo pha..<:.e batch arrival retrial queueing sys::em with Bernoulli vacation schedule,
Appl. i.fath. Comput., 188 No.2 (2007}, 1455. [4] B.S. Dhillon and Z. Li, Stochastic analysis of a syscem omtaining ,l\"-redundaI1t robots and l11-
redundant built-in safety units, Int. J. Perform. Engg., 1 No.2 {2005), 179-185. [5] A. Ehsani, A.M. Ranjbar, A. Jafari and M. Fotuhi-Firuxabad, Reliability evaluation of deregulated
electric power systems for planning applications, Reliab. Engg. S_r•st. Safety, 93, No. 1473-1484.
16 J S.M. Gupta, N-policy queueing system with finite source and warm spares, OPSEA..RCH, 36 No. 3, (1999), 189-217.
[7] R.P Hughes, A new approach to common cause failure. Reiich. Engg. 17 ~19B7 211-236.
[8] M. Jain, N-policy for redundant repairable system ·~,ith acidfrioa rer;Birrruo0n 40, No.2, (2003), 97-114.
[9] M. Jain and A. Mishra, Multi-sta.,,,i:re degraded ;1,ith common cause shock failure and state dependent rates, Raj. Acad. Phy. Sci., 5 ~o.3. 252-262.
[10] M. Jain, K.PS. Baghel and M. Jadow-n, A repai:rab2e system wich spares, state dependent rates and additional repairman, Proc. o{Thirty Fourth Anr.uai Com•entfon of ORSI, t2001J, 134-136.
! 11 j M. Jain, G.C. Sharma and M. Singh, A flexible mariufacwring system with common cause failure, Int.,!. Engg., 1,15, No.I (2002! 49-56.
i 12] D.J. Kim and S.A. Moon, RandoIT'ized control of T policy for an M•o.:1 system. Compui. ln.d. Engg.. 51 No.4 (2006), 684-692.
l 13] K.H. Wang, T.Y. Wang and WL. Pearn, Maximum entropy &.'lalysis to the N-policy M/GJl queueing system with server breakdowns and general startup rimes, /lppl. Maih. 165 No. 1, (2005), 45-61.
Jnanahha, Vol. 37, 2007 fDedicated to Honour Professor S.P Singh on his 70 1h Birthday)
ON GENERATING RELATIONSHIPS FOR FOX'S H-FUNCTION AND MULTIVARIABLE H-FUNCTION
By
B.8. Jaimini and Hemlata Saxena Government Postgraduate Coilege, Kota-324001, Rajasthan, India
E-mail : bbjaimini _ [email protected] E-mail : hemlata_19_750:Yahoo.co.in
!Received: April 16, 2007)
ABSTRACT In this paper we establish some new results on bilinear, bilateral and
multilateral generating relationship for Fox's H-function and multivariable Hfunction. Some kno\\"TI results for the Fox's H-function and multivariable Hfunction are also obtained as special cases of our main findings. 2000 Mathematics Subject Classification : 33C99, 33C90, 33C45. Keywords : Bilateral generating functions; Fox's H-function; Multivariable H
; Generating Function Relationships; Combinatorial identities. I. Introduction. Chen and Shrivastava [1] gave a family oflinear, bilateral
and multilateral generating functions involving the sequence
by
~!.1 __ 01(· ·)= F ~,1; .. _z "' r-c: , ... ,a11 ; t1(p;l - /-. - k ), 131 ,. .. ,I\,; Z)
where for convenience, ~(p;i.) abbreviates the array of pparameters
-~---
i. i.-,-l f . ..,-p-l E 1Y = .iVc
p p p
and for its multivariable extension defined by l p.172, equation (5.21)).
, " A!k1 .... , k_ l k k ""'·- z} "'\"" - •·z· z· , ... ~vr,,c.l, ... , r_ = . L.. ( ~ J '! l ·~· r.
k, .... ,k,=0 1- 1- - P. JK z~
(K =k1cr1 ~ ... +kr<:Jr;k1 EN0 ;1.,crj EC;j=l, ... ,r)
defined
... (1)
... (2)
where {A(k1 ,. .. ,k,)} is a suitably bounded multiple sequence of complex numbers
and (!. )k denotes the Pochhammer symbol.
= (k=O;J,:;t=O)
+l). .. {Jdk-1) (kEN;'AEC) ... (3)
Raina[4] derived the following combinatorial identity as a special case of formula
40
in ([4], p. 187,equation (15)).
f(lc+k-l l[µ+k-
f;;'1l k )\_ k
-L 0. + k-F,
µ+k;
+k,µ-o:; .i· z z
k
= (1-zt\ < ... <Al
n) r(n+l) where = ( ) (. ) k! rh+lfn+k+l ... (5l
Recently in an earlier paper Jaimini et al. [3] generalized the results of above cited paper [l]. They proved six theorems on the generating function relationships in view of the above results
The Fox's H-function defind and represented in the following manner {[2], p. 408), See also ([5),p 265, equation 1,
where
H IL!l
p.q
l -= ~ l q/,:)7:d:
Li ITL ., L · \ - - ~
m n .
nr(bj -f3j';)CTr(1-aj +aj;) <j>{:;) = j=l j=l
q ( ') . f . n r 1-bj + f3.1~ n f\a -o: ~) .i~m-1 .J ~ rl- l
... (6)
71
The multivariable H-function defined a.rid represented in the following manner f[6],pp.251-252, equations (C.
H(z1,. .. ,.<:,.]
= HO.n:m 1 .!l 1 : -~:mr .Hr
JJ.(1:JJ1 -<Jt ~- ··Pr(/,
7 -,.
P ~ 11¥1 I 11 l 1i,
:a~1- , ••• ,a'~ : tc~/ , ·r~t} .n.(li r~•r!). . /d(I! 81 1j , JJ.1 ' ... ' t-1 J 1 (J • \. J ' j
. . ( (rl .,Ir i , ... , cJ , ; J
(d(ri -(r) ; ... ; ) ,OJ
1 = (2ni y t. · t <!>1 (s1 ) ... <j> r (s,. )\p(sl , ... ,sJz;' .. .z: dsl .... ds,;
where i = .J(-1);
ill .II.
n, r(dtil - sfi 1s In r(l-ctil + )ils ·) . j ( J I Jl l . j ' j I
<P1k)= i=l jd . · 7LE
nq, r(l-d(i) + b(i)S·) np, f/c(i) _ )iJs ) . l } I. V; '} I
... ,r}
j=m1 +1 j=n,~l
... (8)
... (9)
' r nr 1-a -'- Ia'.;)s; ..... s_)= = =·
Il r a j=n-....-l
-"'\a:·s ,:_ 1~~
a ,. nr l-bj + If3t;)si / .,! - - }"'°l
41
... (10)
this paper some generating relations for Fox's H-function and multivariable H-function defined in { 6! and (8) respectively are established by
the above cited v.·ork of Jamini et al. (3]. The importance of these results lies in the fact that they provide the extensions of the results due to Srivastava and Raina [ and also provide a \vide range of bilinear, bilateral mixed multilateral generating functions for simpler hypergeometric polynomials. 2. Main Bilateral Generating Relationship Involving Fox's H-Function
Result -1 . Corresponding to an identically nonvanishing function Q~ (z 1 , ••• ,zJ of
s compiex variables z, .... ,z, (s-::: .N) and of (complex) order S, let. - ...
; ~ a.,J2,__,.i:(z,, ... ,z,)ik Hr.s+l j(l-1.-mk-amh,s), 7 7 ., - ..., . . . ... y
,-:, .... -, .. - L { ·.\!' u+l,u ~1 i( ,=· .mk,. . 1. d~,
:;::: O;h E 1Y0 ; S,v E (11)
and
[1; m1 = ."'\ .• 4;··"··"·''( ·t~ ~i+n+amk
L ".11.111 y, Ji u'"' n .;..c;mk-l
·"'·2~ .•J
z 1 ~···,z::; h=O 1, n - nzJ:z n-mk
akD:;-,-J; {z1 , ... ,z, ' ' . J ~.ml< Y.rn -
... (12)
\vhere
]
J~ - n p, t" :--t =I ~-: _ .. · H'·s-1,
l=O(µ-'-n-amk),!l~l u+L·i .,;. , l L
- . i · 11 -1, - n -amk,s), i(c11 , ·r 11 Ji!
5(d () )I· i ... (13) l Ul U Jj
then
-r
I (.y; Z1 ,~.~~Zs; IJ:::;i}
=(l-tt' t m 1 y ~ I
(l-tr ;zp···,zs;(l-tf'Hl)m J ... (14)
Result-2. Let
42
r (. l)mh Q ( \.,k "( /2) [yz z ·t]= "'\' - ah .'i+pk Z1,··•,Zs.Jl Hr,s ii cu,~tu !-1,r'.nz ' 1, ... , :·n L- u,v Y!s, .... . {
kcO :,(d,,0 1 )1
and
. . [nm] _ Nil,11,1 .. G.p,u[ ·z --n]= "'\' u1 .• G.,L.i..l{_.)I"
n,m y, l•···,zs,·1 L... k.n.m \ ' k=O .
~t + n -'- amk - 1 - l a + n + amk -
n-mk n-mk
{ l)mk O ( \.,.,k - ak--&+pk Z1,. .. ,z, 1•1
(mk)(n -mk~
where
ui.,G,µ,u(y;t)= ~ (µ-a)zt1
. Hr,s+l. k,n,m tQ(µ+n+crmkMZ~ u~l.L~l ,c\.
I. - n - amk - l,
Then
J.
"'\' Ma.p.i cr,,t.•l ( . . )tn _ (1-L n,m y,z1, ... ,zs,T1 -n~u
Y T)fm . z 7 • ---'----
(1 - t Y:' l•···•-.;•(1-tfG-l)m
Result-3
Let (2) [ ] . d fi ed. 'l,...' d Y ll,p,m y;z1 , ... ,Z5 ;t is e n rn l u} an
. [n/m] . T3,p,t-,cr,µ,u [y· Z Z . n) = "'\' y1.,cr,.w,µ,a
n,ni ' 1, ... , S''' L... krn,rn
where
k=O
(-irk akQ&+pk (z1 , ... ,Zs Mk (n-mk)!
"' (µ - a)1 t1
H'·s+1 V}:;;;;"11.a (y; t) = ~ {µ + n + crmkUl)! u-i.t·-1
then
µ-n-'-
n-
a+n-'-amk
n-mk
I. -n-amk-rnk-- )l ( ·. 1 , o, . 1, .- 1. - crmR -
- ritm :;: . . . [ ] n ( )-(u-1).,(2) Y ·z ..... z~:. , . • . LTn3,~,1.,cr,o+,u y;zl, ... ,zs;11 t = 1-t IS,p,m (1-t)s' 1- . ~-(1-tJK<hlkm-C<J
n=O
Result-4 Let
I r
y"") [ . ·t]- '(-l)mk n ( ) ·S •m,rr,8,p.«i,i. y,zl, .. .,zs, - L... ak~~&+r'k Z1,. .. ,zs. r k=O
... (15)
... (16)
... (17)
... (19)
... {20)
and.
where
- I~ -n;k ~ amk ~ OJk,st {k,, Yu )}l ,oJr.(-1_ -anik-(c,J-r-m)k,s) J
f?: [y·:z ..... z.:nJ,:::: ') w1 .. ,v . .c .. ·•,p,u(.}''t)
, 1.· , ,s•'I ~ P.'1.m '
k=O
\ -l. . ' )mk ( Lk µ-'-n+amk-1 a+n+crmk-l \-1 akn.':l+pk z1 , ... ,z.'1'1
n-mk n-mk (n-mk)!
43
... (22)
... (23)
w....- i~,'V~N~J.1,U k,n,m
=I (u-a)1ti Hr,s+l r ,,1(l-lc-n-0mk-wk-l,s),{(c",y")}l l=O (µ-'- n + crmk)z(l}! u.,.l,v.,.\Y {(dv,8u}}, (-1..-amk-(w + m )k,s) J (Z4)
Then
x 'e:!:!,r.i-,cr.N .. µ.u 7. .,. . n I:::: n-L..t n.,:n --z. ~···'!-s ~ '.,_ 1
nd)
Proof of Result-I
Y . . ritm 1 (l-t)<='zl, ... ,z,,;(1-t)(<Hl)m+"' j ... (25)
j
We denote the left hand side of the assertion (14) of Result-1 by H[x,y,t]
then we use to the definitons in (12) and (13) we have :
" !n ( ) r -1 H[x,y,t]= L I µ-a it . ~L+n+amk-l a+n+crmk-11J=ll h=ll (µ + n-'- crmk }z(l}l n - mk n - mk
akns+pk (2 1 ,. .. ,zs
(mk)(n - mk ;1) H'"s:1
u~i~L·
-i.-n -mnk-l,s), - )1 ,b". ,j
)11 , Yu f It".
J
Now using the definition of Fox·s H-function from(6) and changing the order of summation and integration and then on making series rearrangement therein, it takes the following from:
H{x,y,t]=~ f$(:;)y; f I . (µ-a)1t1
(µ+n+mk+0mk-1r1
2m L n.l=Ok=O(µ.,... n + mk + crmkMZ)!\ n ; /
k l ' Q (z z ) z -Lk n+mkjd'f-i a+ n+mk+ am - i ak S+pk 1, ... , s f(A +n+ mk + amk + + El;J'I t -, I n } (mk)l {n)! \
44
Now in view of the relation
r(p + n + l) _ ( . ) p + n - I - P• n ! n! · n ... (26
and then i'.nterpreting the inner series into Gauss' hypergeometric function 2F 1 '"'e have:
' 7 " - 1 k . , 1 • k I 1 -~ "\': 1. -'- n + mR _,_ c;m ,.. _._ s:; - µ + n """ m + c;1nR -~: n~ n n
a + n + mk + crmk -1
n
akD.:-J-,,k (2 1 , ... ,z,,
(mk)!
~ i. + n-'- mk - cmk + s~. ~1 -2F1 ..
~L -7-- n ~
+mk+amk-s~ d~
Now using the combinatorial identity ( 4.1 and then on interpreting the resulting contour into H-function \Vith the helf of (6), we atonce arrive at the desired result in (14).
Similarly the proof of Results-2,3,4,would run parallel to that of Result-Ii which we have already detailed above fairly adequately.
3. Some generating relationship involving H-function of seceral variables. The Results-5,6,7,8 given are established multi variable H-function defined in (8) by follmving the corresponding result o:roved in section-2. Result-5
Let vl5 l ['\} ] ~ akQ, · \z- 7 ' :i,1•,ni~0 ,i. J 1 :· . . ,y,. ~ 21 , .. . ,z.-...; t = ) ~ ,,-iv::- l , ... ,- ~
;';:'a (mk) !
Y1(l-
H~~1~~,:~,·'.~.':::''.~-~q. '-
(ai:a~u,. .. ,aj !1". '~; - Jc - crmk - mk; f, 1 , ..• , s r ), (~b,. nili 13lr;
. }'Pj , ... ~ J
and
[n!m] .
R il,f),/,,a,;t,a[ . . ]- '\' B'--cr,µ,a n,m Y11---,YnZI,···1 2 s> 11 - L.., k,n,m
k=O
, ... ,yr;z1 , .... z .. :t] µ + n-'- c;mk-1 - ' ,')' n-mk
4 ')
--'- n - crmk- l \ a;J:2.::,-.1k (21 ,. .. ,zs) k ' ' ' ' ) 1l
n-mk (mk)~(n-mh ! ' ... (28)
where
B~--V.,H .. 0
f;Jl.}1.;;
l •' ,. (u-a zL , ... ,y,;t) = ~ (µ + ~ + vmk)1 (l~
! y. ' •· I i(l . I l } : I -1.-n--a1ni- ;si, ... ,s,.,
~::;:!J
t ~ !
,, ../' ........ j r ••• ,a.: t (b. ;;
..1''
Then
:r
.<11~ .. :p,..</, i. \
i J
I
"\:"' Rr-,p.i.J:r,µ,u [ , .,, . _ . ]tn L 11.m Y1····•J1·•"·;, ... ,Z,,T) · n=O
=fl-V Y]fm l
:::...!.:___.7 7 • I , ... ,(1-t)':• ,-1•··-,-s, {1-t)(c;+l)m J
Resu.lt-6
Let
and
where
,, . -. 7 . ]- ~ (- ynk ( )tk O,i::u, .c·, •... :ict,.u, , ... ,J,-,"-u··.,-s,t - L 1 akQ&+;)k z1,.-.,zs H .... . ·n p.q.pl ,q, ·----r--re'l,
[Y1 ,. .. ,yr;zl , .. .,zs;
a + n + crmk -1 'j (-
·· -mk 1
k=O
y .._, f
,. .. ,y ·z. z. l' ,u+n+crmk-l"-1 r' .a, ••• , s'll
\ n-mk J:=U
ai:Q~_ 0;; (z"' ... ,zs) T)k
(n-mkX
... (29)
... (30)
... (31)
... (32)
z ( ) t r Yi I( 1 k l· ) .,v, ' :l v ·t == L r H ' ' . l' 1, ... , r' r • I E ; ~µ.a(. . ') "\:"' 11 -a zt Ov-'-I·u u· ·u u l. -i~-n-crm - ,sl, .. .,Er,
k,n,m 1'··.,~n · l=0
(u+n+crmk)1(l) p+I,q+I:p1 ,q1 ; ••• ;p,,q, . J(-A.-crmk-mk;t:
1,. .. ,t:r),
Yr
46
(, , (I) (r)) ·{' (1) a, ,a; , ... ,a;. . .cJ. ,
J ·l,p
(b .R(I) R(rl) . (d(li 511) J'i"-'J , •.• ,f-'.1 l.q - . J ' J
Then
'J.
'S ~.p.i.,rr,;t,<I [ Ln L- 11,111 Y1, ... ,yr;z1, ... ,zr;YJ}' 11=0
f = (1- ttti-ll I i Yi
~ 21 ,---~Zr.; . (1-t)1
""11
Result-7. Let 'Y~~!s,p[y1 , ••• ,yr;z1 , ... ,zr;t] is defined in
and
{n1mj • U 9,p,i,,r,,,.,.,u,« [ . . n]- ' Fi .. c,e;.p.;.<
11.111 Y1 , ... ,y,-,z1 , ... ,z~, ~f - L... k_n.r1~ , ... , y_: t] . ,
/::::{)
( µ + n + cnnk-1 'f1( µ+ n + crmk-l {-irk akn!:Hk (z1 , ... ,z")
\ n-mk ) l n-mk ; (n-mk)
where
" {t.t -a.)1t1 ir.<•<.ll.<1[ ·t]=' ;')'ill F·· · · Y1, ... ,y,., L-(,
1..;..n-"'CJmR .• hr-h ,n ,m • /dJ r · · ~ '
r I
o,v+l:ul,vl,. . .,ur,vr I ... n I :
-A-n-crJTi.k-t)R-L;t:1 , ... s,. ,•a j'
Hp+ 1,q + 1: pl,ql , .. .,p,.,qr I
l - ·A-wk- crmk - mk; s1 ,. .. s J,. b1; ~{1 r. . J
Then ,..
"""u:i.p.i ,l'>,!'.ll [ • - • 1.,_n L 11.111 Y1, ... ,y,. .. ~1, ... ,z,, 111 n:::.O
_ 1 )-(i.+l)_ (6) I Yl Yr Tjl111
- . . 7 7 --l t Yms , .. ., ,-1 .. ··.,-,. .. . · ... pl(l-tf' '(1-t)'-· . ·'(1-ti(h!~IH<j·
!
... (33)
...(34)
... (36)
... (37)
.. ,\8) ,· . - . ,· Result-8. Let 1 m,H,p,i.,ci,(•)Lyl ,. . .,y r' 2 1 , .. ., 4 S' i J
~ . k { '~ ~ (-lfi.aknS+ pkl2 1····,zs)
k=O
and
' v;nere
Then
Y1 '1 · k r k k· ) a ·r/ j c1(r)) · v: 1.~-1.-m "-,•) -crm ,s 1 , ... ,s,. , 1 ,,_,_ , .... , ·j l.p .
. a ..... p .. a : . · ' b k·c- c ) [/b . Rlrl " · .. .1-1.-mR-(•)r..,-crm ,01 , .. .,0,. , J, ... ,f-J; y,.' \ J l.q
. ·(c''i .. ~ ... ~ \. ./ ~ f _j
. · l'dj,-) - ,-, ... , j '0
m} . , • _ • ]- ~ 01.,G,<'l,~l,(l [ • t]
, ... ,) ,. , 2 i , ... ,.<:.s' Tl - L.. f;,11.m YI , ... ,yl"' k~O
/ 1 -1 ~l + n + amn - 1 · , k-1\ ( \,.Ii
a.-r-n-ram j(-1YnkakQ&+0k z1, ... ,zs1'1 n-
_____ p,.q.
~ ... '
n-mk :
r ' ) [ _ ~ \~t-a ,t - e·:0 (u+n ... amk)1(lft
(1- /. - n - (1lk - amk-l;s1 , ... ,Er ),(a j ;a j1l,
(-/. -amk- C•)k-mk;s1, ... ,sr ),ib.i;
·1 , '
47
... (38)
... (39)
~6 ' ... (40)
,, ~ v ~4.p.t .. Ci.(")"'·'-' L- 11.111.
n=O , ... ~ yrr; 21 _,-•q z~
i y, I v • I -- .
Yr . . T]tm l = (1- t
m_;1.f'., .. c- 1l (1- , ... , (1- tr- 'Z1 , ... ,Zs' (1- t )l'>+l)m+•» J. . .. (41)
4. Special Cases. If Results- 1 to 5 and in Result- 8 we take
0;1_ 0 1;(z1 , ... ,z,.}~ l ,cr=O and ~t =a these results reduce to the respective known
result in ([7],pp.37-44, equations (1.10), (1.14), (3.3), (5.3), (6.9), (6.6) at ~=0).
If in the result of sections-2, 3 we take cr=O, apd Os+pk(z1' ... ,z.)~lthen
48
these result are reduced into certain families of new generating functions associated with the Fox's H-function and multivariable H-function, but \Ve skip the results here.
All the results of sections 2 and 3, the product of the essentially arbitary cofficien ts
ak ;e. O(k E 1V0
J
and the identically nonvanishing function
Q8+pk(z1, ... ,z5 Xk E ;p,s E .N;& EC)
can indeed be notationally into one set of essentially arbitrary (and identically
non vanishing) coefficients depending on the order & and one, tvw or more variables.
In view to applying such results as in section 2 above to derive bilateral generating relationship involving Fox's H-function and as in section 3 to derive mixed multilateral generating relationships invohing muifrrn:riable H-fu11ction. We find
it to be ccnvenient to specialize ak and Qi: (z,_ , ... , z, l individually as well as separately.
-Our general res·J.lts asserted by sections 2 and 3 can be shown to yield various families of bilateral and mixed multilateral generating relations for the specific function generated in these families but there are not recorded due to lack of space.
REFERENCE 11 ! M.P Chen and H.M. Srivastava, Orthogonality relations and generating functions for Jacobi
polynomials and related hypergeometric functions, l'i{;pl .. \-foth_ , 68 i l995l. 153-188. l 21 C. Fox, The G-and H -functions as symmetrical Fourier kernels. T ron.s:-A..ni.er. Jfai.h Sec ... 98,;1961
395-429. i 3] B.B. Jaimini, H. Nagar and H.l\·LSrivastm·a, Certain da:o-..."Cs of
single and multiple hypergeometric functions. A.dz·.Stcni. 142.
e:c"<Lior.i.s ass:eeiaterl \\iz:h 131-
l4] R.K. Raina, On a reduction forrnu!a involving combinar:D".°ial: ci:H;fficient.s and hypergeometric functions, Boll Un. Math. Ital. A(Ser.7) 4 \1990 .. 1'3;3-189.
[5] H.M. 2rivastava and R. Panda, Some bilateral generating funcwns for a class of generalized hypergeometric polynomials, J. Reine Angew. };foth 283;284(1976 !; 265-274.
loJ H.M. Srivastava, K.C. Gupta and S.P Goyal, The H-functions of One and Two Variables with Applications, South Asian Publishers, New Delhi, 1982, 251-252.
[ 7] H.M. Srivastava and R.K. Raina,New generating functions for certain polynomial systems a..."SOCiated
with the H-function, Hokkaido Math. J., 10 (1981), 34-45.
49
Jnanabha, Vol. 37, 2007 (Dedicated to Honour Professor S.P Singh on his 701h Birthday)
ON DECO:MPOSITIONS OF PROJECTIVE CURVATURE TENSOR IN CONFORMAL FINSLER SPACE
By
C.K. Mishra and Gautam Lodhi Deapartment of Mathematics and Statistics, Dr. R.M.L. Avadh University
Faizabad-224001, Uttar Pradesh, India
E-mai[ :chayankumarmishraca.yahoo.com, lodhi_gautam(g:Tediffmail.com
1 ReceiL·ed : April 20, 2007 j
ABSTRACT M.S. Knebelman [l] has developed conformal geometry or generalised metric
spaces. The projective tensor and curvature tensors in conformal Finisler space
were discuS-ssed by Mishra 1J2][3]). The decomposition ofrecurrent curvature tensor
1n an space submetric class were discussed by M. Gamma [6]. The
decomposition
Sinha and Singh
recurrent curvature tensor in Finsler manifold was studied by
Singh and Garnto [9] have also studied the decomposition of
cun-ature ten:...;;or recurrent conformal Finsler space. The purpose of the present
paper is to decompose the Projective curvature tensor and study the identities
satisfied by projective curvature tensor in conformal Finsler space.
2000 Mathematics Subject Classification : 53B40
Keywords : Finler space, Projective curvature tens, Conformal Finsler Space.
1. Introduction. Let us considered two distinct metric functions F(x,i)
and F(x,i) be defined over an n-dimentional space F~, both of which satiesfy the
requisite conditiions a Finsler space. Th ec.orresponding two metric tensor and
gJJ(x,.i:) and g'Y resulting these functions are called conformal. If there
exists a factor of proportionality between two metric tensors, Knebelman has proved
that the factor of proportionality between them is at most a point function. Thus
we have
where
g'J (x,x) = e-2" giJ (x, x)
gJJ (x,i:) = e-zcr g'1 (x, x)
cr = cr(x)
F(x,i:) = e" F(x,i:)
... (1)
... (2)
... (3)
... (4)
50
The space equiped with quantities F(x,i) and g(x,x) etc is called a conformal
Finsler space, it is denoted by F" .
B ij ( . ) - 1 p2 ij . i . j XX-- g -XX
' 2 ' ... !5
where BiJ are homogeneous of the second degree in there directional arguements.
The following geometric entities of the conformal Finsler space are given
by [7] and (8].
(; ·(x,x)= G'(x,x)-amg"'(x,x) ... (6)
G_jk(x,i) = Gj;,(x,x)- ;Bim (x,i: }um'
-, ( . ). i ( . ' Gjklnx,x = Gjhh _x,x)- ,Bi"'(x,:i.:pm J ·.
B''( . . ). lF2 ;,- _,.; ., x, x = - g - - x· x· . 2 '
... (8)
where G5kh(x,i:) are the Berwald's connection coefficients. They satisfy
i ,c;,(x,x; = G)k·
and the functions BY are homogeneous of the degree in three direcitonal
arguments.
2. Identities Satisfied by the Conformally Changed Projective
Curvature Tensor. The tensor l1T/ and w~.'. transform under the conformal n :r.:.~
change as follow [2].
W;=W;- 112Bim_(.;-,Bim\ .,. __ l_51!0BP"'_l~BP"'') -r~_ xr h h cr"'L h on hrlx n-1 hf' (pl \Gp lrix J n2-l
-~ ·J 1\1:.. B pm ) ( ' 1\(:.. B pm ~~n- NP (h)- n,- J\Ch . ·?(. ?)Br"'GP (' ?)''h "Bpm
""7""'....., .n -- rph - ,n- ....... x {Pl h
( 1 ') ' . · r J ~ Bini ,...i .; B prn n - -.. · t.; ;,,, Born ~ +a (·)x 1oh ---ohc · --n--xc'o · ·-
m i l . n -1 - P nL. -1 n P ?Bw' 2n-1 ....,; - -··.,,·····-
n"'-1
{
. 2 . 2 • · i-': pm i pm X :.. nm . L sm :.. x cPB +am(p) --6hB ---chB· ~a,,,art2B c
n-l n-l ~Bir _ (t" ,Bsm b Bi" .. . le f s
') 1
- 118~ ~Bsma_isBpr -l}"Bsm f ,Bpr i+ ~i:sl {(n+ l)(8lpBsm ~hl(JsBpr n- - n -
...;._ c E,H"r ;], ... ( 11)
= lV£1z - - nx~l ~ [? i =-: rm 'pp ) , 1 -i I +\c,kB 1 , ~-r--2-o1-~i(ri+l) - ' 1,rp J n -1 ,, -
:cpBpr: 9B'"'Gv "i 2 + ,... idrp jj+ CT nd c-Bim n
hi - -'} n- -1
1EDBpm_ XS /31;_-fnBpm ' - n+l - ,_,
2 + ---- CT _; ( -n2 -1 m(p)6fk·tiPchJ(JrBrm -
+ 2G ::-:-Cir ,"'- -Bs"' :.. __ :.Bir -~a· :.. :.. BP,- l _l_;;;i f(c-- Bsm _ _ c .. c. h'c O. ? v 1-\ hi " "' , - 1 ' P , I 1 " l, n+ ,n-
2 a BPT p ·s -_;;. BP''t: Bs"' +~Bsm:.. __ :.. °: Bpr tc' - --° - n -1 c "'c Pc-' ... (12)
RB. Mishra have introduced to obtain the conformal transformation or
projective curvature tensor 1-17) 1d, by differentiating ( 12) with respect to
:i::' ;__ ----r
n+l -.l ~Q PB pm
-c \C_;,B'-"
i;Bp· t;:~p·
Bpm p
6' n~l-Eh (EPBP"'
-:-~>~ n-;-l'
,;;.. ~ch,Bpnt
5s J
n+l
. /· pm) - nc j \C PB •(h)J
-c\i{c:pBP"' ~J+ 2o-m1k hJ\gm) 2 _, noikf (-;, '"'Bpm-) (-a.'"' BPm)}
n2-ll°.m(h)JC/)p -O"m(p) johl _
-"" 2o-ma,[2J 1~ -t,B"' - n-: 1 {c-);!k (a PBsm ~ h/3,Bpr +al!, (a PB""~ j (ah 1E,B1" )
+a_ .Bsm')S . .? '-- Bpr +(\-Bsrn)3 ~,,("':.. Bpr)l- Sj {"' (;, Bsm\;., .tJ Bpr( " . r:t~Pc, - ctr. - /·n;.cpos _J n+l op_clh Pki s j
52
!i' + :z-~\ {(a )31i1Bsm ~ pasBpr -(3 ):PBsm ~hl3sB_vr-'- (3;,:B'·n \i_,C;pasBpr
_, ( ~ sm \~ ~ ; pr l ' oik - cPB f)JOhJcsB )-i--2--
n -1
. (" Bsm \;.;., :., ~ _Bpr {;., B-'il'' + c _1 Y h !f pc ~ - I,( p
.I~ Bsm b :., Bpr - "- I" B,"' '1::: , .. C, 1--r;Cs C 1: \C_,, L SB pr
r :C\Bpr <8lli k J lf;,ftBsm f p<",Bpin" -1
- " '> (" B'"' 1" " Bpr C ;C;, ! CD P1Cs + !•;., B"" \'- . ;., .:, Bpr _ ;., J '; Bsm \:., _:., .:, Bpr Cl ;:'JC_,,c, C;,1\C p O_,C1C5
.:, ("' Bsm _b (" ;., Bpr )- ;., (:., Bsm +cJ c1 l-'hJ cpcs _ c_;pp
_ ('"' Bsm \; .:, ;., .:, B pr op PJOhJOzOs
X U!k f :., '; '; pm 2 -1,,i (
+-9--'(0m(i)( j (hi( PB n- -1
26/c \"'·
n 2 -1
CTm(p{/';,
I"':., BP (.:,BS!" 13 .c!cs P: .~ c/·sBpr
c,_.!(; PBP'" (a ,Bpm )j
Multiply (13) by g1"" we get
-w:ikhgi11 = e1(T giu wjkh + 2e 2'7 umgiu [alk B'r D.~' - ·m
1 ~i -3 . (c. · B' )(h 1: - n-'- 1 ' ./., !!?. '
151 r- (" ) r - '\ J ':"\ -.. pm -. vr m +--v;P o1kB 1111 -\c1kB· )S\,_r n+ 1 \- , I • ' ' "!-'
61 :;;
n2 -1
+a, ,la Bpm 1, \ J -3h1k Bpm
- p ,_ r ~ i: -c ~c
f -i - (• ·. ') x ;., 1:. ...... ...., !,n ____ r·l(
+2e2ag;u(crmlk(hicJB · n-rlcr PBp-rr:: S'.:, c:;)t
n-.l
+ n(~,h I . ' . u (~ -n 2 - 1 I m(h II c c BP!H . J J1 a 12 .? BP"'
ndp)\ J.._,·hi ') ·Jc- f.:, ~e- -g,,,um CY,- le.;
·I .
BPf7' ~;:;
(13)
X ):., :., (°' Bsm\:., "Bpr. '.:') /:., Bsm'i:., ;., Bpr '- {:. Bsm ---1 )('J·clk cp FhJc, +ctk\,cp r- i ics - c; ,CrJ; n+ \- . . - . - ,c,pf;sBpr
+ (1"'1hBsm }"/1i1k/"sBpr )}- nbl l -~ p((\hB;r" };. _,BP,. n;s;.
~R --)-
n- -1
-(3/\,Bsm ~h 1c\Bpr +(8hiB"" ~)':j3,Bpr -v~PB'm ~/:1z/3,Bpr
;. Bsm \.:., ;. BP' _;Ch] r pc s
·I
~2;!k ~hl(tJ.Bsm ~pf;sBpr -CJh1(E
0B'112 0 .3
5Bpr -'-((; ,Bsm E~,c .. BP'-
n -1 \- · · · · J · -·
-(3Pgmi ~[ha jasBpr }+ x~iS[kl [a )ahiaiBsm f /sBpr -313h1(8 PBsm ~J:'5BP,. n -
where
B
I.
~ ~;: G_,::lr ~L
f
P)a B~,. -c, le PB-'" f,
BP' 12 B"' ~
:;C; l.~.B-'"'" f-u
(:, .:. BP'" c~,c ., · X G "' ~ ,,
w =g,,,Hr'
We have the follov.ing indentities.
;., Bor . ;., (-;., Bsm le,,' -:--C 1 C1
!:, :, B",.) (\(_'p{ .-..: ,- .
:, "Bor (" Bsm\;.,:.,:., ;C pcs · - op P/'1z:C B pr l]
' I
,((:.BPm'j1. . J I
! ;:_BP"' l'l] .,l ! i
53
(14)
(15)
Theorem 1. V.7hen F., and F.. are in conformal correspondence, we have
J' -20 :..~\'iBzni -( .(c:, Bim
cJ' k
. x' r. , ..,..----:-nl 'r'i1c"1 !(" BP"',)
T \ L ''· p 'Ud]
.i· ' k'.Bprr J t'' }ij}
0 --.-_---- (
-_i
:, Bpm ). -(" (/: BP"' . ihl] P · h I
-\t Bpr bm ' h I hipr
!:, Bmi: .) :, ('-· pm). :, (:, ""' i/1c,\c1,, · ,, .-ne;,,c1,B 11 +c;, 1c 1
,B· j ' l I '~ n l f j < ] J • f
-i" . .:. (" B"'m X C;JCI i _c 1 '
·.i -1 1 (" Bpm 1 -x c 1jc 111 op
b:1i n 2 -1
,la -Bpm _r,' fi.
-n + tJt, Bi"" +( 12 Bl"'' 1 ... \ p /;. Bpm
-1 r).
- w!: 1
-20. ~c ~<" B' --( ;,e: n
2 .; c\.(8 Bpm) ---1 "'' p ' n+l
5; ,;_f ; , [/., I .
-n-.,-'n- -1
-2G -'- .:. .:. (.:. B pm ) J ' ;., I'- B pm ) --( ' ( ----( (
1 J -~ p ' 2 1 kl p -n_;_i n -
- :. Ir" J ,Bu" 11T1 Jc
.--! -i . • '). i · (. · ) Of/ · (· 'j 'j_ __ J_' _ "-. - B pnt _ __, _ -. . -: B pnt . ·) • ,_ 1 r n ·) 1 ,1 1 v n--1" } · n--1 _,,. · ·
n 51 J, · I· + /na . __ ,,_. - -: i-: Bpm ~ m\p) ·) C li!lC h
8{1, . ~Ci· n- -1 ._1
)l [ ( \ ' pn1 ; ; sni .=... .; tr
1B .12ama,. cu 011B .k31i1c8 B -
+
'n- -1 · ·
;(' {, . . . '. '' - . - (', }3 . -- snr -: .-., pr -._ --... snl -.,. -.. -.. nr ..... sn? -.. ..... ""' pr k cPB c;, 1c,B -c:h\.cpB ~'jici.c,B' +81j ,ohB . h(Jpo,B
n+l
.B'.111 ~h }
((: gor '.\ p
;s: ;J !;..,,
-.,- ·c n- -l ! _:;
,B51" }3._a Bpr -8 (a Bsr" \~- a Bpr l
, • 1. s p k Pnl s J
54
,, - ~t1z
1 {n3.J 1akBsm ~i.,Bpr -ni)1\?PB"m}?
n -Bpr-"- r " "BP,. · ,:( n( s
- n(/3 J!E1:B·"" fJ PE-'Bpr + c\ (E ;;B0''' f,i,Bv -3 )C:PB''" f j/~,.BP"
+tJJIBsm(Jk(J/3sBpr -(EPBsm ~kC:j:EsBpr -i1Ej . . ! :., :., (:., B-"m Y:., :., B pr -r-xcjiokcp .'1PiC.,. :/ 2)3,H'"' ~ .
"" \ ' .t ) i
.[:.. (" Bsm\:.. :. :... Bpr -+ x ('j] .cz r 1<1 ·pc, (/ ~Bsn
!" :/(:..
. (."' B-',,, \; :, :, "' BP,. -X GP Y--'_;iC1:C1C, ie B'm •, p
(': B"m ,c; c Bpr) p s
B pr i- ·) ·\ ("' Bsm . .X C .. CP
.;.-'i_:...Bsm \:. ~... 1.{ :' r -pt'sBpr
t-_Bv )- n\i f, ;,Bsm ~ ,/3PC: _Bpr . . ! •• _; ~ s
- "; (':., B-'171 \; '! Bpr _I:.. B'm Clh CP Y::.11Cs \CiJ ,C:JBF--iE B f
' F ~ "BP"
x1811z(apB"" \:0i"a1asBP,. -x1a, )3,,,la:B""' }3, c Bp· -.1:. (: 1c: fJ JJ l-< ~n" . ,
I o~ .:,_ • . ---0 I
T 2cr ml u) I n 2 -1 h.
l
BP"' p
81 [ /; .
n2 .
1ck1,a_ B-°"'
- p
·) 5;, -'-~~CT•:, n- -1 ""Pich
; . ;, ~ . JP~ :j1 k::: tr c B· l'J ,.. • ... ! s ,J
BP"' ji
8' ·2·/ I/· i - - B''" '1 - ,_ -B"°" 11 16 - x -_,-' - /CT,,,,,-,c ,[ ; (I) !- Ci, I c \C ,,.C' rl ..J' "' l ' \ ' J < , f ,._; n -Proof. Interchange the indices k and h in (13) a..n.d subtracting the equation thus
obtained from (13) and using the symmetric property of the
the result (16).
, we get
Theorem 2. When F,, and F,, are in conformal correspondence, we have
W -W = e20 (W -W . )+ 9e20 c; a, :. -B'" V::.,. - :. · B'" ;ukh .1hlw .1uklz .Jkiw · ~ mb1[:L_ r. IJ.~;_-.r ::.
-a (a B,-m J. k
' :.., (.; . ,c. c ,B'm J \ h1
j:' I·
,- n-l (Jck!(;PBPm c J
(Ci Bpm .P
?/-+ J L {· n2 -1 'PP \CkBpm ~ ("' -c/J .ch1Bpm ~ Bpr \nm , .; BP' 'Y,.m
ck .J .. JhJpr7"Ch] ,1-Ikpr
+~l.;/· n2 -l ('10 j\DpBpm l] - n\a (;.,BPI" J f2j
- c. , I(; -Bpm 1'1J' ·_;
+a"'a Bpm ni\ p
.[;.., {;,, ("' Bpm \ -l; .; ("' Bpm -x oi\uh] Ot ~p)+x ojohl .op
-z 0 hJ f .:, --ino· n 2 -l · .1
; (; Bpm\ '.:, {;,, Bpm') :. ('- Bpm - no j\o P l(k) 'o k \u i 1(p l] - ck ,c P
T·
+x'c
. / "' ;., (a·. Bpm) l+ 9 2cr r . l" ( . .; Bim +x ojok P (l)f, ,,_,e gi[ulC>m(h)(hJ c; x :..
--(;
n+l ..
tBpm
(a BPm) . p i
55
8'_ - J :.. i-
n2_1 ci,;lc~ ;;BP"' : :. (·:... Bom )l J:... I;" Bim .)
n-_:2 lcj cp . 1-CTm(h)ff"h cj n -
x' - . - n-'-l 8i"k(a BP"' , .. ·. ,D
5' __ J_
·? n- -1
(:... BP"'-)-n~ :.(°i Bpm cP . 2
1c 1 ,c P
n -8/1 n -? n- -1
G
-aj
'
-c'
~ .:..._ 1 -. !.' \CI- IBpln
I.:. r:. 1ino'", ·) [· (-. \; . -
!~ +2e-"gilu0mCT,. Ci t,.Bs.r:1 PhJcsB1r n~ -1 · · · ' ' ·
B ir x' };'.: ~ 1'.:.: Bsm b ;,, Bpr ;_, :... /:...- Bsm \o, ;,, BPT s - n+l ~ jck cp Fhlt.s -cjchl\cp f.Jkcs
{8 Bsm .p ) \ ' )" ( ( ;_, pr ;_, :. sm :. . :. :... pr :. · sm .:, · · pr
1csB -ch1(cPB ~)c11osB +cj 8 11 B ~hlapasB
Bsm'1:..:.:. Bor _ {:. Bsm\:... ;_, ;_, /.:.. BPr) (;_, Bsm\:.:.:... (:... Bpr) l ch:cpc, · +;ck P_iC/1]cp\cs 1 - ,ch] pjchCp\Cs ,
+ n5~ l -~~P I.:._ ;: Bpr :. (';_, Bsm);,, ;_, Bpr l+ 8/, f ;, {:... Bsm \.:. :... Bpr C.1z0s -cp,ch Fhlcs 5 ~l lnOj\OhJ ppos
n -
-nc Jc _Bsm vr- . ;.,, ; szn ; .; vr ; sm · ;.,, .:... pr ( \ (" \
0 B· +nch]cpB pjosB· -ncPB ~1chJcsB k'- " B :;,- .:.. 1·.:.. B"'m b cpc.s . -ch] .cp r': sBpr +(8JB"m ~hlhpasBpr
,Ev +x1C':ih 118zBsm f pc\Bpr -i/8 f:. Bsm b,-': Bpr ,C p f- 10 5
. -}.:.. II:. Bsm\.:,:; ;_, Bpr . .z;_, [;_, Bsmb.:,;,, Bpr, -£;,I"' Bsmk:; :. :... Bpr -r-..l.chJ\\OJ P/pCs -xohJ,Op f-!jOzu 8 ,-xcjpl FhJ°pOs
(: )c~ PBsm ,lt:h18/3sBpr + x1(azBsm )3 /3h]a pasBpr - xi (epBsm ~ )h/3/3sBpr f
oh] . .
r.:... Bsm ~ :: Bpr - ;_, I.:.. Bsm f J en pc s nc J ,c p B pr-'- ~ fc:... B"m 'i::,. l Bpr
s . no k.. P f-J Jc s 2 1 n -j_
- j:l C:
,,BS!" P
i(C:JB'9n
,Bpr-'- 12 ,Bsm )~ ,.,f,BlJr -dk(aPBsm ·_\:...Ji70Bpr
- .J . j-1 •. , \ p oJ
BV _/[: ::tBsm ;., BP' , ·; ;., (:. Bsm \; ;_, Bpr Jes -r-XG jok Pl p pas
(.:, Bsm b .:.. Bvr \cp ,.c)zCs . - h Bsm \;., :... .:. Bpr ._1;., (;., Bsm \;_, ;_, ; Bpr
1 Pjopos -x Ok _op pjo1c8
· l .:... .:.... sm :.. :.... .:... .Dr .:... . .:... :c;m ..:... .:.,, :... pr · l · sm ~ .:... .:.... ; pr j ' .. ( ~ + x c JczB ~he pc,B· - c)c PB ,0hciosB + x a1B ' joke pcsB
-xl(aPBsm ·13 )3hf318cBPr ·, f .! ·~ ~ v
9~
2e-v gi[u 8k - . ~nahJ(a Bpm -1 p
8/ii :... . l 2--ch(a Bpm)l n -1 . P J
+ -i ~i 'l f 8i . ; bk ~ .:.. pm b h l :. ;., pm , -I k :... :... ;_, pm
- 2 -c1:1(c jB )--2 -ck(ojB ), --r-X CTm(l) - 2 -ojohl(oPB ) n -1 n -l i ! n -l
) \.
_ bh] B-Bk(BPBpm)~-X1Gm(p) 2 1 J ' . n -
8i ;., 3'i ('; Bpm)' 8~1 :... ; la- BPm)j(l7) -2-CJUh] Gt +-2-ojok~ l . n -l n -1
'i6
Proof. bterchange the indices u and h in 14) and subtracting the equation thus obtained to (14), we obtain the identity (17 .
Theorem 3. When F,, and F11
are in conformal correspondence, we have
W . - W = e20 (w + W , , ;11/w u1k/1 111/;h !{lilt::
O' . x :_ " I'- B 4- b
1:!.~-- c t ,t '1 ;, \{ !' n + 1 ' ,.. ·
'),.-, 4e-·'c;n,
g 0. -~ il ~· (p
B ir hm -1-I1i: -(
B p1" I --• 1Ui.fj
r:B""
\nm PhJpr-
i) l ! . gi1u lk ;nc J·
- n2 -1 . ;Bpm I l-nt
1,i6MBP"'
' ·~P - , f-'
(e; }Bpm \ f
((::BP"' •P
'-, LC .B '- ~B CT [-g J:_ /:_
1c :f-- 1'-- ! \(
.1:{ . ,1"- BPlll I l'/ i. \C' Bpm -c' n+l JI
l p ; ·)_._· (~ n- -1
iZ0B n ·)
n- -1 " p
- 0 m( iV~ )'J ,BP"' p . }· h, 2G ~ 4e CJmurgi!u ci.i
" ,
. " BET - x' "' Jes . 1 a
n-1
(·,'- B'',,,k",," BP!' -(\.I:- B",,,k" 17: '- j) ' I} i( -" if,· \t p ~' ',i..
(" B"111 \:._ " - " B'n- 5·,. -+cik t'j1Ch(pt.,' - 2. ,t.J
n -J.
:., :,_ Bpr -'1 ("i B"m \:_ :., Bpr . £P£" -ncj1Cp 'rids ..,- B''"!2 f.
C- Bw (
c
BP' - PBsm ~ iC Bpr I . s I
. :,_ (." B"n' \:._ " Bpr :,_ f,:, B"'" \'- ,c: - ( h ! ( J i r ;} s - i h ; ".' p r .' l. B .••r
. - . H,,., ~hie B;;r s
( :., B"m \:, :., :., Bur .f:_ '- I'- B"m \:., '- Bpr .:, '- (.'- B''m ~cp ,_.h!rj_;(_., · +xcJ!Ch\_c1 F'pCs -c_;Jch.],cp _,BP''
, ~ ~ sn1 .:.., ~ ~ pr ~ ~ sni ~ .:... ~ pr , .:..., .:... sm .,~ . ~ ( ' ( ' . . .
.,-(fzi.l/B fj,cprsB -chJ,cpB ~jiC/vsB -:-Cj,(clB ,th;/!- _,Bpr
-c:,_j;kpB""12 }\1c"'1c\Bpr + (l'1B"111 ~ hjC B pr (::.· s . - c ~, .:, :_ pr )ij1
lclcsB ,J
+ 4e20 g(u011, [J n -1 Pm(j
(C; Bpm , p crm(p)ch;
Bpm jl f ;J
.;.. ~I I :,_ . ;, 'l sm '_ . j::o . .:, , . :., pm 1 18) . ' 'l] . x Pm(/lc.1J. 1z1c. PB. J CTmlp),,.. J,(onJo!B Ji,.
Proof. Interchanging the indices in each pair (j,u) and ik,hJ in equation 14J and
adding the equations thus obtained from (14), we obtain the 18).
Theorem 4. When F11 and Fn are in conformal correspondence, we have
wiuhh + wjlzlw + wldzju + VV/wjh = e20
(wjhlw + wjukh + whizih + 4 20 g e ()m iw
57
~ -B'rhm ( (~ µhi -c B1m ·/ x
_ -'-(:, ;(C:h1Bim )(kJ] + n + l )) - l j
- (· pm) i' k ! C' PB (h))
i' -- n-"-l~C[J(:h,((: Bpm . L ... \ P
8, - f - ( • ) - ( - - \ -'- n ~ 1 ·p P Ck1Bpm (h))-cli ,~il!Bim l(h))
-Z ,It. Bim _-(' . -i r -
(-'- Bprn) X ;._ ;, (" Bpm \ ' ,- 11 ' 11- n .,..1 ·•c: hiF p -(h)) - n + 1 (':/ hl cP l(k)}
6: - _·-' ~
n ~ ,_. ~ . Bpm
. -1 --
;:.. (0-: Bpm ') i; BP,. ·-\r.111 , (" Bpr \r.m 1 1)-c.p,Lh• -(k)j-\Ckl f.-Ih1pr' c,,) Pldpr)
s; , 1 -1-. ) - (- ) - {- ) tR ;_ -: :::: pnz : ...., pin , ....., ~ pn1
-~1 fCJJchJB (p)-ncJJ oPB (h)).,-oh 1 cJ 1B (p) n -
(,? ~Bpm !-'
(iBPm \ ;-·,
~B";;;m r
n) --. -. pm ~ ; pm 8'- i· - (• ' - . - ) n 2 _1 fC[j cklB )(p) - norJ{c PB (k)]
1 _ t:-:r JC: Bprn
' ;,'• \, p
' :C n 2 - l au(i.i1Bpm
- i x' -
. I -x' 8' lh ;, f- (. n 2 -1 C'Jlfh1 ftBpm \ }(p)
-C:rklaPBpm ~ 51
4 ·)" [ 1:.. (; Bim -:- e- gi111 Cim!k'(C°h!\Cj] -a (; -
n + 1 j I 0 h /!' B pm p J ----c'-
n Tl lli B pm )'.
p )
- Gndh}}
"}
n'" -1
- - - i -i _ (·-': --Bim )-'-_X_; .(-=: ,-1 Bpm \, ~ -':, !.? Bpm 1- ( , 1 • C[ I ( k ,( n f+ CI• I,, n " ' .1 ' · n + 1 - · · ' ,, -· n -:- l ·· · ic
r.,
n.1Bp.u
B pr. p
n 2 -l
n 2 -1 p
BP"')'
B pm ') ("' ;, Bpm )l p ,-CTm(p)OlJCk] j
2rr r;, {;, sm '}:.. ;, ir ;, (;, sm \; - ir ~4e c;m0.rgiiulcu\cklB Ph!csB -G[j\OhiB PklosB
-1 x
n +1 i'1)C PBsm }:-. BP' _ ~ _;, . (-=: Bsm \; · ::i Bpr
•0 s clJchicp Phlcs
.:.... .:.., ..:.., sni .:.., .:..., pr . .:.. ; sm · .:.. ;, pr · ~ sin , , }3 {. ) (' ) -cijch1\opB FkJcsB +c!k(oPB JJ PhJosB -c1i 1 cPB ~ 'i ..'.... pr i ..:... .:.. sm .; · · pr ; · sni :... .:._ :.. pr . ) (. ' ( p -cLi(,ck!csB -r-c[j PklB phJoposB -cfJ akJB "hJaposB
{" Bsm \; :.. (" ;, Bpr .) (" Bsm \; 0- (:., " BP,. )1 - erk p j]Ohi 0 pas +Ph) PrJ kJ\O pos f
. .J J:.. (" Bsm \;: ;, Bpr :.. (" Bsm \;, 2- Bpr l -:- n+l f'p ch, fk1°s -cp cki F'h1 s J
'58
. c)l ii J /.:., '; sm \; '; pr 1.:., .:., .-;m ~ , pr
.-.,--in\cilch!B PpcsB -n(cd-:JJB r111 csB n~-1 · · - ·
. (~ Bsm \;, .:., .:., Bpr (.:., Bsm \::. . .:., .:., BP,. , n c 111 r: j]c pas - n\C 0 Fpc hies . . . . :.. ~ sni .:., pr .; ~ srn ,;.., ;., pr , ;.,, sm ~ .:., ..:.._ pr f (.· .f ! •
-r-(1iJ 11 B pcsB -chicpB . ncsB -,.-(cJ 1B ~h!cpc~B
(.:., Bs111 \.:., .:., .:., Bpr - cl' K-'1i 1Cilcs ')
n- -1 ·Bsm f .:., Bpr ! . pcs
f.:., ':i Bom 1-1 .:., B pr , B"m t :... ;, B pr (" B:;m \ -npuc,; Ph,cs T h . Jlcpc". -n.,cp I
: .... ".'.'l pr ""\ ......, srn --; -. pr , ~ -:- srn ....., .... or -: snz -... """' -. pr • • • • (. • • • • (" ! • • i. • . . .
c[jch]csB +clh cj 1B ~pcsB -rC[k.CpB ,BJlcsB- +\cuB ~-k)cpcsB
-(.:., Bsm \.:., .:., "?, Bpr cv Prhr,,~,
. ! < x·o,;, !.:., __ ,. tc 2 1 .. n -
.atBsm f :')c,BP' - c Bsni i":i Bpr 0 ,_ s
-r th. ;(("1-B'm f ;,/:;Pa _Bpr -
'' , J j .'}
;C ,BJ)' ..;.. c ,-B 0m fh,fc ;/,B
"""' -. Si1'1 ·-. t .-.. ..... ;Jr . l.. '- I - . -cj]cpB Ph!lclcsB· f c B .!:c i:' B·'"' i:'::. ;:; ' ' B s -,cp ,Ll]L-h..JCJC.s
ih5'· J.:., . - \.:, . - --~ru(c;d171Bsm ppcsBpr -cr;3k1\80Bsm
n- -1 · · -J • •
. ~ ;; - L --. B pr . ~ I--. Bsm 15 cs ...,. c[h ,_c i .. Pc,Bpr
..... -. snz ......, .... -. vr .-.. sm -.. ....., -.. or . (. ·~ - - . ~'- . . . -c1kcpB ,111 o 1c 8 B· +oij B , .l.?JlcpcsB-
.:., (''.:., Bsm \; (.:., .:., Bor ·J. C[j _Gp Pl?J.OfCs . I
B sm b .:., .:., .:, Bpr (" Bsm b .:., .:., .:, Bp·· Fuc ldr Po s - c P 1-·uo k;c ic.-; .7,
4e~0 g, _ n2' - I
-Gm(p)ah) (a j]Bpm )+ x1 CT m(l)a j)(ahl(; PB pm
51 , ( . (· ) h1 l "· .., om - -"-- )<J ml 1·)C h CJ /JB · - CJ mi_p n~ -1 · ·· · '
' 18 -Bpm \ .. n
, . . 1· )1 x18!, - I_ .. · J -, .., --, pm . i. f, --. J-, - ~ DT'1
-x CT ( 101 ch' c1B 1--.--c,(·h:lc1B' mp, .J ! .. - .· n2 -1 .J ... , , -
(7 . ~ .m~ C'
~ '._/,_.,
-en
IE:BP·T..
_B 7 '' l
((!)Bpm) tl - l(!j iJ
. 2rr .:., (:., im \ x' .:., +2e g;,Gmlk ch] ojB )---.. cj
n + .i
'j 51, - .. B pm - __ .! - --. • • I --- B pm p , n -i- l C n1 .C p
2n8fk 2 i (. . ·) 1-. . ., ) • --' - () . < • pm - . , ..... ""'' - pnl r ' . :..c r-
T 2 le g1u'(fm(h)]OJ8PB . c;m\pi\C;CnJB !1'2e giuGmur n -
•l . , . . .
:., .:., Bir X )°' .:, {.:., Bsm \:., .:., Bpr , -'-. ('.:, Bsm b {;, .:., Bpr ') Oh]Cs - n+l 'f jO[h\cp PhJOs TC[k _op f-'j\Oh]Cs
( . . ~ ( , · ~ sn1. · ~ ~ pr ; sm ; ..:.., ; .:.... pr -r-8jPlkB f1zJcposB -(clkB jchJ,CposB
8' . l I· -- y: !"' ~~
n 1 ,~ v .C·. B·""
T ' ~ · _ f1
(8,,.Bsm) ' ,._f~ i
Bpr}
· 8 lh ~nfa -8 Bsm \:., a Bpr - n(ao:., Bsm b a Bpr ~ nla, .Bsm T 2 1 ( ~ J h] pp s . J p Fhl s , \ f1•
n -cpcsBpr
59
_B-~m \:.. ~u ,t'
. '-. _Bpr _ ):.. _ .f '- -Bsm \;., ;., Bpr _ ', _ _{:... Bsm \;.,.:.. Bpr _,_ ics '/Cn;·Cj 1t'pos ohj\cp P/'s '
./.,;
Fe Bsm \;., :.. ::-: Bpr (:.. Bsm \;., ;., :.. Bpr I, X b[k L, (" ~ Bsm \;., ;., Bpr -,c P;,icpcs -,cp Prhcjos 1+--z-ltc'J ch1cz Ppc,
(.:.. Bsrn '1:, :.. B pr j C p LfC, . -
" 1·:.. Bsm \;., I :.. ::. B or - c Ct l'."'hl ,_cpc, .
-l8_Bsm \;., , p F
+ _-i;i r. V"-"J
;C:c,BP-'
~Bpm ,...
n -I'- Bsmh;., ;., Bpr ;., (;., Bsmk;.,;., Bpr
•iCJ CjOpC,_. -Oh] Cp 1CjCtCs
:.. I'- Bsm \::_ (;.,;., Bpr) (;., Bsm\::i::. ;., ;., Bpr c·J ,cp ,ch]\c1c_, + c 1 t- Jehl° pcs
gm [k :.. ;., pm ;., ;., pm , 2e2G . bi rr .. ')
fJm(JPhJ(oPB )-crm(p)Oh](ojB )J
"' "' (ia. Bpm)l] crmip)C jch], I f
. .. (19)
Proof. The proof follows in consequence of (14) anad (17). 3. Decomposition of Conformal Projective Curvature Tensor. We
considered the decomposition of conformal projective curvature tensor in the form
=.)Co (20)
\vhere ~ is a homogeneous conformal decomposition tensor and Xi lS a non
zoro •;ector such
X''Vi = l (21)
Similar manner the decomosition of projective curvature tensor WJkh in
the form
l-V)1c1i =Xi qi Jkh
where the decomposition Yecto: ,yi also satisfies the condition
X'\~ = 1
Transvecting (22) by :e, >ve get
where
=X'¢!?h
.t.. . i == -'+' 'hbxJ ...
The decomposition tensor $ ikh satisfies the identity
=
(22)
(23)
(24)
(25}
(26)
We notice that the decomposition vector xi and the recurrence vector v;
60
are trans£'ormed conformally as under :
J(i =e-0X1 (27
and
V =e"V l I
{28l
respectively.
Using equation ( 20 and 22 l in equation ( 13), the obtained equation
transvercted by ~ and using the equation (21), 23), {27) and (28), •ve get
:+: - G ~ ' v 2 f;, Blr 'l'jkh -e 'Vjkh T ; amL01k
;, \i - --o, a .. B'"' _; -.A ---~~ n -'-l ·
1. BDm \ap · i1;,i]
81- f
_} i;, i' -f.J(31k(3111Bpm ti n + 1 '(-'p\_O\kBP"' )} - pr G"~ B .h;pr n 2 -1
f;, 1.ohlBpm
;.,_ (" Bpm) ;., (; Bpm -no.ioP (h)]+chlloj -ah la BP"'
t ., p - i'5[k ? -
9 '-' i n- -1 , (81Bpm
- (} h l (/3 p B pm )(!) }] + V; 20 ml k (: h i (C: j B im y'
n .. 1 a I (c\ .(1 BP112')
-,- • , .. I p
' l '
8j ;, (" pm )I Vin6f 1c ---c,·O B r+---n + 1 · ii P · J n 2 -1
,C DBpn1 f-G J •
' v I;, ( • Bsm \:,, :.._ Bir '• x' _\;., - Bsn: L'.: - B [J.!' -r i20m0,.10;8r1c P111Cs -.---lie C"<; ;,• L-C. , L . n- .
;., ;,, (';, Bsm );, ;, (;, Bsm \;:; ;, (.;, Bsm ~ ;:; B_:;r ;., B·""' \;:._ [" ;., Bpr j'
-cju[k,Op pjo[k\Op f"jo[k,Op 1-'n]Ls -c:k _1--'j\cpcs ,
; (·a· Bsm \; I;, 0· Bpr) :.._ (;, Bsrn \;, :.. :.. Bpr :.. (,,:.. Bsm \;, /;, ;, BP,. -O[k ·p pj~uh] s -C.iPU: :Chlcpcs -Oj,Op PLk\O;zJCs
(·,_ Bsm\;:., ('"' :.,Bpr\(':,, Bsm~:., {;, ;,Bpr)l 8) ):... {;, Bsm\:., :...Bpr -o1h pjchJopos J-CP t"'jc[k\Oh]os J-n+l!Gp\p[h PkJos
n8' 1·(. . ~ . , _ _ ,_ . _ _ ._ lk \t"1 - Bsm - ; Bpr !- - Bsm13 - BP'. I- B.-sm\;:: +-2--- tcjohl pos -pjcp . hies -,ch: ,._
n - .1
Bpr
-(aPBsn ,0/ihlasBpr}+ ;rh ~ ~h 1 (8jBsm "pp(\BP,. -c\1(3PBsm}3.ic\BP,. n -
(. ~ _ _ (· ,_ . . '] Vih;! [,· ...... ::;rn '"' ~ ""'\ pr -i snz ""'l ; pr , r lk ~ - cjB _ h)c:pcsB - cPB ~ihojosB f.,. n 2 -l c j
:., :., ('"' Bsm \:., :., BPI' . :... (:., B"m '\;,, :... ;., Bpr :.. i;., Bsm b -cjc-hl op Plus +oh] u 1 /-'jopos -ch \Op c
,0;, -Bsm 'i:., ;, Bpr
j 'i r pcs
,SB pr
-,.--c
_ 1_;; k
61
1" .i-t;, 12 B pr
' cjlcpB-"" fh Bpr )_._ (~ B"m \.:. _;_,. ~ r~ Bpr s . ' .c l ;= J '- n 1( p - s
1- - - - IJ Xi - r- · ) 0[1k - -· . ; . ~ ~ ~-.. pr .' _ -r-_ ') _ _· _ -: . "':'! --: _ pnz __ ' _ ; . ~. prn
-f.e-,,,Crc,B I V,~G,.,1" CJ C[kC1, 1B + 2 c 1,,c 1 B .) n, • _, - - ... 1.~ 11 -c- l - ' '· n - l J
0 .,, ,ZiBP"' J .. --:-- "\-~5lh 2,-u,,,i, h ' n- -1 -
[i _Bpm p
v -l~i ix ork ! ..
n2k_1
2Pmu/' 1 (2hf Bpm)l - . I p f ·)
n- -1
___ (29)
represents the conformal transformation of the decomposition tensor under the change r l
Thus \Ve state
Theorem 5. Under the decomposition (20), the conformal decomposition tensor
is expressed in the form i29)_
Transve(:-ting equation \20) by _;;,),we have
= x 10:.i
Using the equation (23), (28) and (30) in the equation (12), we obtain
,,B'm xi (r_
il - - ~c _ t-,. BP"' ' n --:--1 P '"'
-'--\ D l 'Phl1pl
{30)
(31)
- l .,.. -;-,-- o~ -1 ,._ B::- ~PB 11 - l/,,, J- BPm 1'111 p ;-)] + 2B G1z111) l n- - 1 !
-'-
+e"Vi
f.::·
r n:B12: _ _ • ~r '
n -l
'J - ,, p
I B pm l
D [ . J
e"V? +~CT -
n- -1 ndp tt:P<\J,.Bm' - hJBpm uvY 2 rr.:. Bsm f.:. ;:, Bir
e i O"mO",-t\r·[k "('h(.s
- t x -- --;;-1 i'\ ,(-- c~ B pr 1 (, -r j p .s ~ -7- - sr
I 71 -1 Ufk h]Bsm » i SB pr -(2 lz]2 SB pr}.:. PBsm
2 --:- -Bsm:.. - _ n -1 'h ,1-, r Bpr , ,. .. } s
(32)
Interchange the indices k and h in the equation (32), we get
=-~hk (33)
62
by vritue or relation Wkh = -iv:i:k [ 4].
Li the veiw of the equations (23) and (24), the equation (32), becomes I
J. CT,i_ av2 i(;., Bim '+'kh =e '+'kh -e i Gm! 0[k
L i'
i]- n -c-1 pafkBprn ,Erm QP,
R 0 hJp
_._ 1 ~i J . . nz -1 oi1: in+ l~c 111Bpm i -n(3PBpm
_(.~ ;., ,
l] ,~hfpBpm ll B rmap. r + 2 k1rp- .d
+ecr~2crmi(h)i B im n s:.i ;., Bpm i' ;., ;., Bpm , ----v"•C ---oh.a j n 2 -l ''' P n+l '' P
"TT 9 e •,~ ..,.-2--
n -1
'
ii:P c"';, if Erm -, -~" r
x' ;., ;., ;., BPrl 1 _, ---C111CnC~ ,--Op,
n + 1 ' " ·· 'n -1 ·"'
' 2 Bsm '-: 0., ;., B pr ,-- C-111C µC,
n-1 · ·
.B·:J."'''-
1B"m l~P(: ,BPT - ';., BP'"'§ B"m JCS , p
J;, ;, ; f';,jc,B'r
Which gives the conformal transformation of decomposition tensor
under the conformal change. Theorem 6. Under the decomposition
tensor ¢ hh is expressed in the form 1 34).
' a.no
REFERENCES
the conformal decomposition
1 i M .S. Knebel man, Conformal geometry of generalized metric spaces. Proc. ~Vat A.cad. Sci. US..A,
15 (1929), 376-379.
! 21 R.B. Mishra, Projective tensor in conformal Finsler space, Bull. De Ia Classes des Science, Acad
Royale de Belgique. f1967J, 1275-1279.
131 R.B. Mishra, The Bianchi identitites satisfied by curvature tensors in a conformal Finsler space. Tensor NS., 18 ( 1967), 187-190.
! 41 H. Rund, The Differentiai Geometry o{Finsler space. Springer-Verlage 1.1959;,
15 i B. B. Sinha and S.P. Singh. recurrent Finsler space of second order H Rep. Indian J. Pure and
Applied Math. 4 No. 1, (1973l. 45-50.
161 M. Gamma, On the decomposition of the recurrent tensor in an Areal space of submetric class, ! 'l'T. of Hakkaido Univ. (Section) 28 (1978), 77-80.
! 71 HJ' Pantle, Various commutational formula in conformal Finsler space, Progress a{ Mathematics (2i, ~ (1969J, 228-232.
! 8 l A.K Kumar, The effect of conformal change over some inti ties in finsler space, Atti delta Accad.
Nax Linceli Rendiconti, (1-2), LII, H972l, 60-70.
19 i S.P. Singh and J.K. Gatoto, On the Decomposition of curvature Tensor in recurrent Conformal Finsler space, Istanbul Uni. Fen. Mat. Der. 60 (2001l, 73-83.
Jnanabha, Vol. 37, 2001 (Dedicated to Honour Professor S.P Singh on his 701
h Birthday)
EXTENSION OF P-TRNSFORM TO A CLASS OF GENERALISED FUNCTION
By M.M.P. Singh and Neeraj kumar
Department of Mathmatics, Ranchi University, Ranchi-834 008, Jharkhand, India
(Received: December 22, 2006; Revised: May 10, 2007)
ABSTRACT P-transform is a new transform, which is defined on O<t< x:. Testing
function space P0 has been defined, so that the kernal function of the transform
belongs to ~,, . Properties of P5 have been studied. P-transform has been extended
to a class of generalized function. P-transform has been shown as a particular
case of convolution transform. A real inversion formula has been derived for P
transform. 2000 Mathematics Subject Classification :44-02; Keywords : P-transform, generalised function, testing function space, inversion formula.
1. Introduction. Let us consider a transform
{1. P[f(t)] = F(s) = ~ ("' 6 t
5
,J(t "yi,t (0 < Re(s) <co), 1! • 0 t + s
where fit} is a suitably :restricted conventional function defined on the positive
real line O<t< x and O <Res< x. The above transform maps f(t) into a complex
valued function F{s).
Testing Function space P6 and its Dual P' l! •
Let Pr; be the space of all complex valued smooth (infinitely differentiable) fucction
$(t) defind on the positive real line O<t< x: s.t. for each f(t)E Pfl.
Then
(1.2) Pk(+)= supj(l+tfDk4'(t~(~~l) 1J<t<Z 1 ·
(k=O, 1,2 ... ).
Hence, Pk is a norm on P13 (k=O, 1,2.,.) and {pk};=0 is a multinorm on Pl>.
64
Thus, ~1 is a countably multinormed space Zemanian ([3], pp 8-9).
Here ?-Transform is a new transform defined on O<t< x. Testing function space
~'has been defined so that the kernel function of the transform belongs to Pf\ .
In the p1:3sent paper, we study the properties of Pn and extend ?-transform to a
class of generalized function. P-transform is shov.m as a particular case of
convolution transform. Finally a real inversion formula is derived for P-transform.
2. Lemmas. In this section, \Ve establish four Lemmas.
Lemma 1. P,; is a complete space.
Proof. To prove that P is a complete space. It is sufficient to prove that every
Cauchy sequence in Pf. is a convergent sequence h'l.
Therefore, for every m,n>Mk (a fL~ed positive enteger)
- a sma'J +ue E s.t. < s.
Consequently, vve get
(2.1) ') k. { . ~ +tY D'¢ 11 t)-$m(t1i<s :;
But 3 a smooth function <jl(t) s.t. for each t: and i.
Due to Apostol [l,P, 402], we get
(2.2) + t yi D"$ 11 (t )-$(t 1i<s. Therefore, as n ~ Cf)
P;,i<iln -¢!--+ 0 for each k (k=0,1,2 .. .).
Also, since $11 (t) E P13 , so we get
(2 3) l(l+t)13D;,th (t'M<C. .. . .. I \f'n ~ k--
<;'.<__:( .. fl>
as rn. ~ ::c.
where Ck is a constant not dependindg upon n. An appeal to (2.2) and (2.3} ghres,
+ ty" Dk¢(t ~= !(1 +t )f1 Dll¢(t )-$,, (t )+ ¢12
(t) ' '
:S ;(l-,- t ¥' D'11 ¢(t ~- ¢,, (t)+ l(l + t )!\Dk ¢12 (t )< s +Ck (from 2.2 a.nd ' ! ' .
which sLJWS that the lim ... c function ¢(t) E P~,
Hence, {¢,,}is a convergent sequence in P1,.
65
Therefore, is a complete countably multinormed space.
Let P' be the dual of P,. Then f E P',, iff it is a continuous and linear functional
on P0 since is a testing function space (Zemanian [3],pp 38, 391}.
So, P' is the space of generalised functions.
Thus, for any f. E P'ri and o E P,5, value of the generalised function is denoted by <
t: CJ>.
Lemma 2. To prove (2/;t) t5 /(t6 + S 6) E ~' for 13::::; 1, 0 < t <co and O <Res< w.
Proof. Let us consider,
2 ,..., ;;A ( ~6..:. t6 )_ 6,iO i(t6-'- ~6·l2 ; ;;.~vl \::i , I l f ' ;:;; ~ •
Consequently, we get
(1..:. +ss = P,~ ) Qh (J):,; 1),
~'irhere are the polynomials in t s.t. the order of Qk(t)>order of
Therefore, we get :
T rr )t5 \ -'- t 6 ),! bounded for all 13::::; 1,0 < s <co and k =0,1,2 ...
Hence, (2 t5 . 6) ~ n +S ""-'Ir·
Lemma 3. is a subspace of ~: , ;vhere has been defined [3, pp. 8-9].
Proof. Let. $ E l is bounded,
where f(t) is a complex valued smooth function non zero within the compact set K of I= ]O, x (and zero outside K.
:::::> +t Y' Dt:$(t}, is bounded for all p::::; 1, k = 0,1, ... 1
:::::> tj>EP13 .
Therefore, we get
{2.4) D(I):,; ~,.
Thus, the convergence of a sequence in D(l) implies the convergence of the sequence
in P13. Consequently the restriction of Pµ to D(l) is in D(I).
66
However, D(l) is not dense in Pi>. Thus, we cannot identify P0 -..vith a subspace of
D'(l). Actually, the different members of ~~ can be found whose restriction t-0 D'{l)
are identical.
Lemma 4. Pf\ is a dense subspace of E(l) where E(l) has been defined [3, pp. 8-9]
Proof. Let 9 E Pi, => sup Dkq;(t) is bounded, where !) ::; 1, k::::. 0,1,2 ... i},h
=> sup1Dk9(t '1 is bounded where k is a compact set of I =[0, x] !Ek . 1
=><PE E(J).
Therefore, we get
(2,5) Pf\ <;;;; E(I).
An appeal to (2.4) and (2.5) gives
(2.6) DU! ;;;:; Pii ;;;:; EUi.
Also, DU· is a dense subspace of E(l). [Zemainan (3.P.3.7].
Thus, from (2.6) it follows that Pr, is a dense subspace of
result.
Hence, we get the
3. Extension of the P-Transform to a Class of Generalised Function. Let us call fas a P-transformable generalised function if it possesses the
following properties :
i. f is a functional on some domain d((l of conventional functions.
ii. f is additive in the sense that if e,qi,e + 9 are all members of d(fJ, then
<_ f, e + <P > =< f, e > + < f, 9 > .
ni. Pit :;::: d(f) and the restriction of f to Pf! is in P' c •
Also 2/ n. t5 ;(t6 + 8 6 )E Pµ for 13::;; 1;0 <Res< x.
We define the generalised P-transform of f by
(3.1) F(s) =P[f(t)]=<f(t), (2/rc).t5 (t6 +s6 ) >
where, s E 0.f ; and
(3.2) 0.f = {s: 0 < Re(s)< w}.
Thus, f is called the region of definition of P-transform and 0, x are the abscissa
67
of definition.1'.i:forever, we call to the operation P:{~F as P-Transform.
4. P-tr.ansform as a Particular Case of Convolution Transform. Let us consider the convolutional transform
(4. H(x)= h(y)G(x- y )dy(- Ye< x < 'i))
and let us choose for the kernel
(4.2) G(x- y)= j[ J 1 _ eSIX-:.
and G (t) = (2 rr 1 .,.. e6r .1.
Let us change the variables of i_4.1) and (4.2) by putting s=eX and t=eY. Thus we get
t4.3) H{logs) = ~ f ,-_, h(logt) t5i(t6+s6) dt n: .-,_
= r fdlogt) t5 +s6 dt (O<Re (s) < 'i) ). ~;;:
If we put HOog s)=F\s) and t)=f(t), then (4,3) becomes
2 +s6 0 < Re(s) < ::n), =
~'t
which is identical \vith n. Therefore, (1.1) is a particular case of convolution Transform.
5. A Real Inversion Formula for p-Transform.
Let llE(s)j;;: = G(t'Je-stdt •-:r.
2 ,.,. l =
1+
= ~ -z 1 I .s-1 , n: .o l+x-6.:x ax substituting = x)
1 2 f. xs-l ___ I dx Therefore, we get E(s) - n Jo
2
TE
5 s
~dx x6 +l
s+5 21~· x_dx =- 6 rr l+x
\~ --, T-
68
(5.1) 1 ~- 1
E(s) - 3Sin[(s + 6)/ 6]rr => E(s) = 3sin( S-'- 6 . \ 6
(i) if (s+5J is an even positive Integer. tiiJ (s+5) <6 i.e. s < l.
zs+5
The result (5.1) is obtained by the contour integration of --6 on a semi 1+ 7
circle with centre at the origin and radius R---.+ J). Therefore, E(s)=3Sin[(s+6)/6J ;r.
where E(s) is called the inversion function corresponding to G(t).
Nc:w the inversion formula of (1.1) is given by {2p.8}
3 Sin [(D+6l/6];: H(t)=h(t (- x < t < x).
The substitution t = logs=> H(log s) =
Therefore, (5.1) gives
(5.2) 3sin[(D + 6 )! 6 ]rrF(s) =
which in an inversion formula of ( 1.1).
REFERENCES
=> lltiOf!S*=
(0 < s <
l ll T.M. Apostol, Mathematical Analysis, Addition \Vesley, Reading, Mass, 1957
[2] I.I. Hirchmann, The Convolution Transform, Princeton Cniversity Press, nm1::eton.1'ie'l!i
1955.
[3] A.H. Zemanian, Generalized Integral Trorr.sform.atwn, inter Scien'.::e Sons, Inc, New York, 1968.
and
!4] M.M.P. Singh, An Extension of Q-tra.risform w Generaliize>::i Report, 18 No. 1&2 <1999 & 2000J.
The Nepali Math. Sc.
69
Jnanabha. 37, 2007 (Dedicated lo Honour Professor S.P Singh on his 701
1i Birthday)
ON APPROXIMATION OF A Fl!'NCTION BY GENERALIZED NORLUND
MEA..NS OF ITS FOURIER SERIES By
Ashuthosh Pahak, Murtaza Turabi School of Studies in Mathematics, Vikram University, Ujjain , Madhya Pradesh
and Tushar Kant Jhala
Department of Mathematics, Government Postgraduate College, Mandsaur-
458001, Madhya Pradesh 458001, India
rReceived: July 17, 2007)
ABSTRACT
The aim this paper is to establish a theorem on approximation of a
function generalized Norlund means of its Fourier series which generalize
several previous results.
Keywords and Phrases : Fourier series, Generalized Norlund means.
2000 Mathematics Subject Classification : Primary 42A24; Secondary 42B08.
1. Introduction. In 1943, Iyenger [7] proved a theorem on harmonic
summability of Fourier series: The result of Iyenger [7] was generalized by Hardy
Hirokawa [5], Hirokawa and Kayashima [6], Pati [10], Prasad [14], Pandey
Rajagopal [15], Siddiqui (16] and Singh [17], for (N,pn) sun1mabiliity of Fourier
series under different conditions. Dealing with Cesaro-means of Fourier series of
a function, Flett [2] has obtained a result on the degree of approximation. This
result was generalized by Izumi, Sato and Sunouchi [8] and Siddiqui [16] by using
Norlund means. Working in the same direction, the result of Siddiqui [16] has
been extended by Ponvru [ Gupta and Pandey (3] and Chourasia [1]. The purpose
of this paper is to establish a very general result than those of Porwal [12], Gupta
& Pandey l3J and Chourasia [l] so that their results come out as particu.lar cases.
2. Definitions and Notations. L€t f be 211-periodic and integrable in the Lebesgue sense. The Fourier series associated with fat a point x is given by
au ++(a_, cos nx + b,, sin nx) 2 ~ ~ .
where an and fr,, are Fourier coefficients off and are determined by
an=_!_ i"J(x)cosnxdx rt --,{
(n ;:o: o)
and
70
b, =.!.f"·t(x)sinnxdx Jr -TI
(n>O).
Let (pn) and (qn) be sequences of positive constants such that
11 n n
p" = LPk, Qn = Lqk, Rn= LPkqn-k :;:': 0 (n;:: 0) h=O k=i} k=O
where Pn, Qn and R_--+ o::: as n-'Tx.
"' L La" be series whose nth partial sum is denoted by Sn.
n=G
Write
l " f·i;:·q = R LPkqn-kSn-k·
n k=iJ
OJ;
If N :·q -'T s as n -'To:::, then the series Lan or the sequence (Sn) is said to n=O
be summable to S by generalized Norlund method. We ¥.'rite for each real x,
Ht)= f(x + t )+ f(x - t )-2f(x)
and 1 or [l/t], the integral part of in 0 < t :S. :r.
3. Main Theorem. If 0 ::;.; a. ::;.; 1, 0 < 8 ::; ;: and x is a point :
. R( <D(t)= J''j$(u~~du
t u
=O[{R(l;u)h(t)}a] as t-+0,
where
{R(l/u)h(t )}a -+ oo as t-+ 0 ,
f ~ {R[u Jh(u )}a du = o[t{R[11 t Jh(t )}a]
and h(t) is a positive increasing function such that
h(t)-+ 0 as t--+ O, then
N f:·q (f;x)- f(x) = ol(Rn t-1hfiin]j+0[1/ Rn].
... (3.1}
.•. (3.2)
... (3.3)
4. Lemmas. We shall require the following lemmas in the proof of the theorem:
71
Lemm.a 1. Let the sequence (pn) be non-negative and non-decreasing, then for
0 s a < b < .YJ and for any n,
uniformly in o < t ::;; i!, where i'-'f is some positive constant not necessarily the same
at each occurrence. For its proof see :i\k Fadden [9].
Lemm.a 2. Let the sequences (pn) and ) be non-decreasing, then for uniformly in O<t<n,
b '
L Pnqn-k sin(n-k + 1/2) t) = O(R,).
Proof. We have
. , :-1 n sin(n-k-,- t!=!I+ I sI1+I2
.~=t
where
r-1 I 7-1
I1=IIPnQn-ksin(n-k+l/2)tl sipnq11-k sipnqn-k. k;O hO
Now, since q., 2 qn-i, therefore
q, __ k 2'.q 11 _1: for n2:-r.
Hence
L l $ LPtq:-&:· l;=i}
By Abel's lemma
1 r
Lz Sprmax!Iq"_ksin{n-k+l/2)t rsrsnik=r
where
,J r j~ • ( I iL...q,,_ksm n-k+l;2)t ~!:='
i r ,
=II qn-k ~in(n - k + ,%)tcoskt-cos(n + 1/2)tsinkt} Jh=r
72
$ 1Iq11 _" coskt/ + ltqn-k sin k=: i lhr
s MQ,, by lemma 1.
Thus I 2 s MP,Q, s MR,
since R, = Ipkq:-1: 2 P:Iq:-1, = P:Q: . 11=0 .fr:::;\?
Combinir.g above results we finally have
n t
LPkqn-k sin(n -h + 1/2) ti= O(R,} j • ~
Lemma 3. Let (pn) and tqn) be the sequences as in lemma 2. Then
1 11
2-D )p q sin(n-k. . JUL .._, /1 n-k -,- 1 ?J
n k=O , . - -smt 2 -
Proof. YVe have for 0 :::; t :::; 1/ n
1 I/ - "'"" sin(. k . , 2r.Jl ~ Pkq11-k n - + 1 2)ti
u k=O . ' smt 2 ! = i
O(n}
O(R7 /tRn)
O(lf Rn)
(0 st s l (l/n < t < t'S)
(8StSTE)
t 2
= O(n) 0 :St~ 1. n ~
Now for 1/ n < t < 8
! 1 !l
i2nR LP1;q,,_1. sin(n -h+ 1 2)tl
n h=O . · . i smt 2 i I
1 1· ,, s 2rr:R11 t; P1;q,,_k sin(n ~ k + l 2)t smt 2
=If Rn O(R,/t), by lemma (2) and since sint/2 2 tire (0:::; t:::; :r)
= O(R,/tRn ).
Similarly we can prove the third part of the lemma. Lemma 4. Under the condition (3.1), we have
J~l$(u~du = o[t(R[u]l'- 1(h(t)Y1) as t ~ o.
Proof. We ·write
)= l©(u~ R ,,.;u •' [luf"
u
then by hy-pothesis of theorem, we obtain
= - J~ u<D(u )du
= -[u¢(u )] ~ + J~ <D(u )du
•t ( = -t<D(t h j
0 <D u )du
= O~(R[1 rjh(t)f j+ f~(R[1,ufi(u)f du
r . , ] = Olt!Rr1 • -Ji(t) f , by (3.2)
l ", t . !. j /
Hence, we have
=
=
R; .. 'du ) ;iL u 1
1
R[u]
R~1 ,~
l <P(u) JRr1-11du I
J
i _r _ .1 ~ I tRfl 1 (h (t) )a I i
{l t t q f j'
= O~Rfi~J 1
J ast-+0-
by (4.1)
73
(4.1)
5. Proof of the main theorem. Let S,, (f; x) be the nth partial sums of the
Fourier series off Then (N,p,.,,q,J means of S,,(f;x) is given by
1 n .
N:,'-f;(f;x)= R ~pkqti_hSn-k(f;x) n -~~o
where
Sn(f;x)=! f"{f(x+t)+ f(x-t)}sin(~+12)t dt. rr 0 2smt_/2
74
Hence
1 ft
Nf:·q(f;x)-f(x)= R LPt:q,,_k{S11_1:(f;x)-f(x)} ll 1:=0
1 $(t) n smt 2 LPkqn-ksin(n-k+ll2)tdt
k=O 1 =
2rrRn
n r" f" + i + > .\ !~ • ,s
= I 1 + I2 + I:i (say). (5.1)
Now !J1 =of n J; 1
" 4i (t )dt} by lemma 3
I; 1 ( )a-l(hr \ct J = 0, n- Rn ·11/n]J l' by lemma 4 \ n i
= O((R,Ju-i(htlin]f) · (5.2)
Next, we have
r 1 ,) j$(t ~ iI2! = 01 R J1 .n-t-R[1 ,pt ['by lemma 3
L JI
f 1 . f-1 =DI -\Rnhti. n]J ' by ,Rn
= D((Rn )u-l(ht11n]f ). (5.3)
Finally, we consider
, l 1 Jrrj$(t~ vi 11 jl3j =01-R & -R[IJq~t , by Lemma 3
~ !1 t J
= O(l/Rn) {5.4)
by Reimann Lebesgue theorem and regular conditions of summation procees. Combining (5.1), (5.2), (5.3) and (5.4) we get the proof of theorem.
REFERENCES 111 J. Chourasia, A study in the approximation of function by Fourier E:qxz:n.sion, Ph.D. Thesis, Vikaram
University, Ujjain, M.P., India <1993).
75
[2] T.M. Flett, On the degree of approximation to a function by Cesaro means of its Fourier series, Quart. J. Math. 7 (1996), 85-97.
13] S. Gupta and G.S. Pandey, On the degree of approximation to a function by the Norlund means of its Fourier series, Vikram Math. Journal, VI (1986), 42-48.
L4l G.H. Hardy, On the summabiiity of Fourier series, Proc. London Math. Soc., 12 11913), 365-372.
15 J H. Hirok.awa, On the Norlund summability of Fourier series and its conugate series, Proc. Japan Acad., 44 \1968), 449-451.
!61 H. HirokawaandIKayashima, OnasequenceofFouriercoefficient, Proc.JapanAcad., 50<1974l, 57-62.
f7l KS.K Iyenger, A Tauberian theorem and its application to the convergence ofFoureir series, Proc. Indian Acad. Sci., 18 (1943), 81-87.
!8l M. Sato Izumi and G. Sunouchi, Fourier series XIY, order of approximation of partial sum of Cesaro means, Z. Sue. Tuna. M.JA., 35 <1971), 157-194.
19! L. Mc-Fadden, Absolute Norlund summability, Duke Math. Jour. 9 (1942), 168-207.
10 j T. PatL A generalization of a theorem of Iyenger on the harmonic summability of Foureir series, lndian J. Jfoth. 3 1961 •, 85-90.
[ 11] G,S_ Pa.Tidej~ On~£ Norlund summability of Fourier series, Indian J Pure Appl. Math.,8 (1977 !, 4l2-417.
[12] J.P. Porwa1, A Stud:« in the Theory of Approximation by Special Orthogonal Expansion, Ph.D. Vikram Cjjain, M.P., India (1989).
I 131 K Prasad, Norhmd summability of Foureir series and its conjugate series, Indian J Pure Appl. Math .. 4(2) (1943!, 179-184.
14l CT, Rajagopal, On the Norlund summaility of Fourier series, Proc. Camb. Phil. Soc., 59 (1963), 4'.i'-53.
15 J A.H. Siddiqui, On the degree of approximation to a function by generalized Norlund means of its Fnurier series, 36th Annual Conference, Indian Afath. Soc., Madurai (1970).
116] T. Singh, On Norlund summability of Fourier series and its oonjugate series, Proc. Nat. Inst. Sci.,
India Pan A, 29 fl96.3L 65-73_
Jna.'1.abha, 37, 2007 !Dedicated lo Honour Professor S.P Singh on his 701h Birthday)
A CO:MMON FIXED POINT THEOREM IN MENGER SPACES By
R.C.Dimri and N.S.Gariya Department of Mathematics, H.N.B. Garhwal University
Srinagar-246174 .Garhwal), Uttarakhand, India
E-mail : dimrirc!?1ogmaiLcom
tReceh:ed : Ivfay 3. 2007;
ABSTRACT In the present paper we prove a common fixed point theorem for six mappings
in Menger spaces_ Our result unifies and generalizes some of the previous known theorems due to Dimri and Chandola [2], Sharma [9], Singh and Chauhan [11] by using a different contraction condition. To some extent we replace condition of
R->veak commutafr;,rity of the mappings. AMS Subject Classification (2000) :47H10, 54H25 Keywords: Complete 2\fonger space, R-weakly commuting maps, Common fixed
1. Introduction and Preliminaries. Menger [ 4] introduced the notion of probabilistic metric spaces (P11'1-spaces). Sehgal and Bharucha-Reid [7] initiated the study of fixed points in a subclass of probabilstic metric spaces. They extended the notion contraction and local contractions to the setting of Menger ~paces. The study of this space expanded rapidly with the pioneering works of Schweizer and Sklar[6].
Sessa [8] initiated the tradition of improving commutativity conditions in metrical common fixed point theorems and introduced the notion of weak commutativity. lvfotivated by Sessa [8], Jungck [3] introduced the notion of compatible mappings. In [his connec[ion Pant [5] introduced the notion of R-weak commutativity which asserts that a pair of self-mappings (f,g) on a metric space lX,d) is said to be R-weakly commuting if there exists R>O such that
d(fgx,gfx)::::; R d(fx,gx) for all XE X. Recently, Singh and Tomar [12] presented a brief development of weaker
forms of commuting maps and obtain some results for non-commuting and noncontinuous maps on non-complete metric spaces. In 1998, Dimri and Gairola [l] introduced the concept of R-weak commutativity for a pair of maps in probabilistic metric spaces and established a fixed point theorem for generalized non-linear contraction.
In the present paper we prove a common fixed point theorem for six mappings in Menger spaces by using the notion of R-weak commutativity. Our result unifies and generalizes some results due to Dimri and Chandola [2], Sharma [9], Singh
78
and Chauhan [11] with less restrictive conditions on mappings. Definition 1.1 [7]. A distribution function is a mapping F: R ~ R+ which is nondecreasing and left continuous with inf F=O and sup F= 1. We shall denote D by the set of all distribution functions. Definition 1.2 [7]. A probabilistic metric space is an ordered pair (X,F), where X is an abstract set and Fis a mapping of Xx X into D i.e., F associates a distribution function F(p,q) with every pair (p,q) of points inX. We shall denote the distribution function F(p,q) by Fp,q. The functions Fp,q are assumed to satisfy the following conditions: <PM-1) Fp,q (x) =1 for all x > O iff p=q, (PM-2) Fp,q (0) =0
(PM-3) Fp,q = F q,p ,
(PM-4) If Fp,q (x)= 1 and Fq,r(y)= 1 then Fp,r(x+y)= 1. Definition 1.3 [7]. A triangular norm (briefly, a t -norm) is a function t:[0,1] x [0,1] ~ [0,1] satisfying the following conditions: (i) t(a,l)=a andt(0,0)=0, (ii) f(a,b)=t(b,a),
(iii) tt.c.,d) 2 t(a,b) for ca, 2 d 2 b, (iv) t(t(a,b),c)=t(a,t(b,c)).
Definition 1.4 [6]. A Menger PM-space is triplet (X,F, t) where (X,F) is a PMspace and t-norm t satisfies
F p ,r<x, y) t {F p,q (x) ,F q}Y)} for all x,y 2 0 and p,q,r E X. Note that among a number of possible choices for t,t(a,bl= min{a:,b} o:r
simply "t=min" is the strongest possible universal t Schweizer and Sklar [6] have proved that if (X,F, t) is a Menger PM-space
with a continuous t-norm, then X is a Hausdorff space in the topology T induced by the family of neighbourhoods
{N p(c:,I~): p E X,c: > 0,), > o} where Np{c:)_) =Ix Ex: Fx,p(c:) > 1-
Defination 1.5 [1]. Let A and B be two mappings on probabilistic metric space X. The pair (A,B) will be called R-weakly commuting if and only if
F ABu,BAu(Rx) 2 FBu,Au(x) for all uEX and R>O. Definition 1.6 [7]. A sequence{un} in a probabalistic metric space Xis said to converge to u if and only if for each A. 2 0, x 2 0, there exists a positive integer N(x,A)EN such that
Fu,,,u(x)> 1-'A for all n 2 N.
Or, equivalently !,~.F,,,,,,, (x) = 1.
Lem.ma 1 [ 4]. Let { un} be a sequence in a Menger space X. If there exists a number k E (0,lJ mch that
79
'2F11 (x) for all x>O_and n=l,2, ... , then {un} is a Cauchy
sequence in X. 2. Main Result. In this section, we establish the following common fixed
point. Theorem : Let A,B,S, T, I and J be self mappings of a complete Menger space (X,F, t) ·where t is continuous and satisfies t(x,x) '2 x for every XE[O,l], satisyfing the following conditions: (1.1) ABO[i = ST(XJ c f(Xi 1.2\ if either a) I or AB is continuous, pairs UV3,I) and (ST,J) are R-weakly
commuting or ;_a· J or ST is continuous, pairs (AB, l) and (ST,J) are Rweakly com.muting,
(1.3) F ABu,STu{kx) Flu,Ju(x) for O<k< 1, x>O, u,uEX. (1.4) (A,B) and CS, T) are commuting. Then A.,B,S,T, I and J have a unique common fixed point. Proof: Let be an arbitrary point in X. Since AB(X) c J(X), we can find a point u 1 in X such A.Bu =Ju:.. Also. since ST(X)cf(X) we can choose a point u2 ·with STu 1 =fo2 . using this argument repeatedly one can construct a sequence
} such that = ABu .,,, =JU:p,,,.:.., a..nd
"'"'1 _,,, c ...._
Y<>~~ 1 =STu.Jn..1..: =Iu,)~_,_ 9 for n=0,1,2, ... ""'.;r;;i.. ~ J;,.,_J;,. ' .... -f-1 , .....
Now from(l.3) and properties oft-norm,
-1 (kx) = FsTu2,,.,.AB11..,"." (kx)
= F,1Bu"2c·"2 .ST112n ·l (kx)
'2Ftu" .. ".J''i _(x)
"~~*·:.. r
= F~" ., ) .
In general, Fv_,,,_,(kx);;::Fv ,.·,· - ; - ti; • - n.~ .. ·- >:;
for all n EN.
Thus, by Lemma l,{yn} is a Cauchy sequence in X, Since Xis complete, { y n }coverage to some point z in X.
Since {ABu 2n}, {Juzn+l}, {STuzn+l} and {Iuzn+z} are sub-sequences of {yn}, they also converge to the point z as n-"><D.
Case I. Let us assume that I is continuous. Then the sequences {I2u 2n} and {L4.Bu 2n} converge to Iz. Thus for x>0,A.E(0,l), there exists a positive integer N(x,t.) such that
(1.5) Fz.4Bu..,, ,Jz(x/2)> l-i, and
F12u2n ,Iz(x/2) > 1- /, for all n '2 LV(x),).
80
Using (1.5)-, we have
FAB!u,,,Jz(x) > dFABfu2
, .!AB11, (x.'2),Ff.·1.Bu,, 1zlx 1 2): for all n 2 N.
Using R-weak commutativity of AB and I,we have
F.wiu,,, ,Iz (x) > t{Flu1
,, ,ABu~,. {x / 2R),F1ABu,,,,Iz (x / 2)} .
Therefore from ( 1. 5),
(1.6) ABiu2 " .-+ Iz as n ~:;:..
From(l.3),
]<ABiu.2,,,STZL,,._, (kx);:: F;2,,1 ,ju.2,,-1 (x).
On Lettting n-..+oo and from (1.6), T·Ne get F 12)kx) ::: Flz,z(x), which is not possible, since Fis non-decreasing therefore
lz=z. Again using( 1.3),
F.4Bz.STu1 ..
1 (kt'.):'.:
Letting n ~ oc , we have
F.4Bu (kx) :'.: F1z.z (x) ~ 1 which implies that ABz=z.
Since AB(X)c J(X), there exists a point win X such that JW=ABz=z, so that
STz=S'Tz=ST(Jw). Now from ( 1.3),
Fz.STw(kx) = F.4Bz,Sru-(kx)
;:: Flz,Jw(x)
= F2 ,2 (x)-+ 1, a contradiction. Thus,
(1.7) STw=z=Jw, which shows that u: is the coincidence point of ST and J. Now, using the R-weak commutativity of (ST,J) and from (1.7), for R>O we have
FsTJw,JSTw (Rx):'.: FsTz.ch (x )-+ 1.
Thereforn ST(Jw) = J(STw). Hence S1'z=ST(Jw)=JSTw=Jz, which implies that z is also coincidence point of the pair (ST,J). Using (1.3),
Fz srz (kx) = F AB- sr' (kx) since ABz = z , , -, -
~ Flz,Jz(x)
implying thereby STz=z=Jz. Therefore z is a common fixed point of AB, I, ST and J.
Case II. Now suppose that AB is continuous, so that the sequence ~AB}2 u2 ,, and
L4Biu2 ,, converge to ABz. Since (AB, J) are R-weakly commuting, therefore, . I
.ABz (x) > tWrAB:·
> t·if' 'Bu . . ."I.
for all n::::: N_
AB!:. 2),FAB!u"" ABz(xf 2)}
2R\F_wJu", ·.wz (x 12 }i-
Letting n -+ x , above inequality implies that
(1.8) L4Bu 2 , -+ .4.Bz -
~'\gain from (1.3), we hm.-e
F_413z_z 2: F.1R 2 (x). Therefore ABz=z_
As earlier, there exists w in X such that ABz=z=Jw. Then,
,- (kx ::::: F2AB. -L
which on taking the limit n -+ :x reduces to
81
coincidence STz=Jz.
2:
ST
:chr:" -;mplies that STw=z=Jw. Thus w is the
Since the pair (ST, J) are R-weakly commuting. then
Further, ) ;::: F] -J: (x) reduces to
Fz.STz(k.t:)2:F2 s1z(x) as n-+x, gives STz=z=Jz
Since ST(X) c I(X),there exists a point in X such that Iy=STz=z, then
F AB:u(kx) = FABy_STz(kx) 2 FlyJz (x) = F2 _Jx)-+ 1 which gices ABy=Z _
Also fAB, n are R-weakly commuting, we obtain
I since ly=z=ABy
::::: Fly_AiJ_, . R
= Fz_z (x / R )-+ 1 gives that
ABz=lz=z Thus z is a common fixed point of AB, ST, I and J_
the mapping ST or J is continous instead of AB or I, then the proof that z is a common fixed point of AB, ST, I and J is similar. Let z' be another fixed point of I, J, AB and ST, then
Fz.z {k.-c) = F ABz,STz (kx)
2 Flz.Jz(x)
2 Fu (x) implying thereby
82
z=z' . Hence z is unique. Finally, we prove that z is a unique common fixed point of A, B, S. T; I and J. We have shown that z is a unique common fixed point of A, B, S, T, I and J. Then, using commutativity of A and B, \Ve have Az=A(ABz)=A(BAz)= AB(Az) \1i'hich shows that Az is a fixed point of AB, but z is the uni qi. .e fixed point of AB. Therefore
Az=z=ABz. Similarly Bz=z=ABz. Again using commutativity of S a_nd T and in vie\\' of uniqueness of z, it can be shown that
Sz=z=STz, Tz=z=STz. Hence z is a unique common fixed point of A, B, S, T; I and J. Remark 1. If we put I=J, A=S a_nd B=T=Identity map in the Theorem, we get the following result: Cor<;>llaryl. Let A and J be self maps of a complete Menger space X such that (A,J) are R-weakly cm:nmuting, J is continous and
F.b . ..;,.(kxL~FJ".J"\.xJ for all u,uEX, x>O and O<k<l.
Then A and J have a unique common fixed point.
Remark 2. A number of fixed point theorems may be obtained for two to mappings in metric, probabilistic and fuzzy metric spaces as the special cases from The Theorem.
REFERENCES fl] R.C. Dimri and U.C. Gairola, A fixed point theorem for 2 nodine:air contraction. Gur1L
Kangri Vijn. Pat Aryabha.ta, 12 (1998), 155-16CL [2J R.C. Dimri and VB. Chandola. A Corrunan fi.xed theorem in fuzz:.- metric spaces, J. Nat. Phy.
Sci., 18(1) (2004), 103-114. [3] G.Jungck, Compatible mappings and con1mon fixed points. fotemat. J Math. & Math Sci. 9t9}
(1986), 771-779. [4J K.l~fonger, Statistical metrics, Proc. Nat. A.cad., Sci., GSA 28 \1942), 535-537. [5] R.E Pant, Cornman frxed points of non-conmmting mappings, J. Math Anal. A.ppl., 188{21 ( 1994),
436-440. [6] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland Series in Probability and
Applied Mathematics, 5 (1983}. 17] VM. Sehgal and A.T Bharucha-Reid, Fixed points of contraction mappings on probabi.li:,"i:ic metric
spaces, Math. Systems Theory, 6(1972}, 97-102_ l8J S.Sessa, On a weak commutativity condition of mappings in fLxed point considerations, Puhl. Irr.Si.
Math. (Beogard) 32 (1982) 149-153. [9] S.Sharma, Common fixed point theorems in fuzzy metric spaces, Fuzzy Sets & S:ystew .. s, 127(20021
345-352. [10] S.L. Singh and B.D. pant, Common frxed point theorems in probabilistic metric spaces and
extension to uniform spaces, Honam Math. J., 6 (1984), 1-12. ! 11 l B. Singh and M.S. Chauhan, Common fixed points of compatible maps in fuzzy metric spaces,
Funv Sets & Systems, 115 (2000)471-475. i 121 S.L'. Singh and A. Tomar, Weaker forms of commuting maps and existence offLxed points, J. Korea
Soc. Math. Edu. Ser. B: Pure Appl. Math., 10(3) (2003), 145-161.
83
Jnanabha 37, 2007 (Dedicated ro Honour Professor S.P Singh on his 701h Birthday)
PERFORMA...~CE PREDICTION OF MIXED MULTICO:MPONENTS MACHINING SYSTEM \V"'ITH BALKING, RENEGING, ADDITIONAL
REPAIRMEN BF
J
M.Jain and G.C. Sharma Department of Mathematics, Institute of Basic Science, Khandari Campus,
Dr. B.R. Ambedkar University, Agra-282002, Uttar Pradesh, India E-mail : [email protected],[email protected]
and [email protected] and
Supriya Maheshwari Department of Mathematics,
St. John"s College, Dr. B.R. Ambedkar University, Agra, 282002, India
(Received: July 18, 2007)
ABSTRACT The present investigation deals with a stochastic model for multi
components system with spares and state dependent rates. We study the machining
system where failed units may balk with probability ( 1- f3) and renege according to
exponential distribution. The queue size distribution in equilib1ium state for the system having M operating units along \vith mixed spares of which Sare warm and Y are cold, is established using product type method. Since the reliability of the system, depends upon the system configuration, the provision of r special additional repairmen turn on according to a threshold rule depending upon the number of failed units the system, is also made. The expressions for some
performance measures are prmrided. The expressions for expected total cost per
unit time has also been facilitated. By setting appropriate parameters some special
cases are deduced which tally with earlier existing results.
Keywords : Machine repair, Mixed spares, Queue Size, Balking, Reneging, Additional repairmen, State dependent retes. 2000 Mathematics Subject Classification: Primary 90B50; Secondary: 91B06.
1. Introduction. The study of fascinating area of machine repair
problems via queueing theory approach can play a crucial role in predicting system descriptions of manufacturing and production systems. In the industrial world, the machine repair problems arise in many areas such as production system, computer network, communication system, distribution system, etc. When a
84
machine fails in a system, the interference occurs in the production if all spares have. exhausted. A machine interference problem is said to be Markovian, if the inter arrival time and service time are both exponential. Machine repair problem with spares and additional repairmen is an extension of machine interference problem. In view of machine interferenc, the provision of standby units is recommended as by using these units the system may keep working to provide the desired grade of service all the time. There are three t:ypes of standbys namely cold, warm and hot as defined below. A sta .. ndby machine is said to be a cold sta..ndby when its failure rate is zero i.e. only operating units fail. In case of warm standby~ the failure rate of spare unit is non zero but less than the failure rate of an operating machine, and is called hot standby when its failure rate is equal to an operating machine. The available standby unit may replace the failed unit \vhenever the operating unit fails. The behavior of the failed unit depends upon the number of failed units ahead of it.
For maintaining continuous magnitude of the production, it is recommended that the spare part support and additional removable repairman will be provided. It is worthwhile to have a glance on some of the relevant works done in this area. The que~eing modeling of machine repair problem with spares and /or additional repairmen has been done by many researchers. Gross et al. ( 1977) considered the
birth-death processes to study markovian finite population model i;.'fith the provision
of spare machines. Gupta (1997) introduced machine interference problem \\'1th
warm spares, server vacations and exhaustive senice. Jain 1998) developed model for M/M/R machine repair problem with spares and additional :repairman. Jain et al. (2000) and Shawky (2000) studied a problem one additional repairman
case of long queue of failed units.
The concept of balking and reneging for machine repair problems in different framewo::-ks was also employed by many researchers \Vorking in the field of queueing theory. Jn recent past, queueing problems with balking and reneging have been studied by Abou El Ata (1991), Abou El Ata and Hariri (1992) and many others. Jain and Premlata (1994), investigated M/lvf!R machine repair problem with
reneging and spares. Shawky (1997) and Jain & Singh (2002 considered machine
interference model with balking reneging and additional servers for longer queue.
Ke and Wang (1999) developed cost analysis of the M/M!R machine repair problem
with balking, reneging and server breakdowns. Jain et al. (2003) investigated a queueing model of machining system with balking, reneging, additional repairman
-and two modes of failure. Al- seedy (2004) presented a queueing model \Vith fixed
and variable channel considering balking and reneging concepts. Jain et al. (2005) suggesteµ a loss and delay model for queueing problem \vith discouregement and addition::J servers.
85
paper, we study machine repair problem with balking, reneging,
spares and additional repairmen by using birth death process. The spares are of
two types namely cold and warm. The life times and repair times of the units are
assumed to be exponentially distributed. The terminology of the model and
notations used are given in section 2. In section 3, the governing equations in
steady state and their product type solution are provided. In section 4, some
performance measures are derived. In section 5 cost analysis is made. The discussion
and idea5 for further extension of the work done are given in the last section 6. 2. MODEL DESCRIPTION A1\1D NOTATIONS Consider mfaed multi-components machining system with balking, reneging,
spares and additional repainman. For formulating the model mathematically, the
folloVving assumptions are made :
* There are lv1 operating, S warm standby and Y cold standby units in the
system.
*
*
'l'
*
*
*
*
The system work with at least m operating units where for normal
functioning M units are required.
The life time and repair time of units are assumed to be exponentially
'The repair facility consists of C permanent repairmen and r additional
removable repairmen to maintain the amount of prodution up to a desired
goat If the number of failed units is more than the permanent repairmen
then we employ the additional removable repairmen one by one depending
upon work load.
After repair, the unit will join the standby group. \Vhin an operating unit
fails, it is replaced by cold standby unit if available. If all cold standbys are
exhausted, then it is :repiaced by warm standby unit.
The repairmen :repair the failed units in FCFS fashion.
We assume that 13(0:::;; 13 :::;; l) is the probability of the unit to join the queue
when all permanent repairmen are busy and some standby units are available
and B, when all standby are exhausted and no additional repairman turns
on. When j (1:::;; j :::; r) additional repairmen are working, the balking probability
is given by 1- f3 j .
Failed units renege exponentially with parameter u when all permanent
repairmen are busy and standby units are available. In case when all standby
unnits are exhausted and number off ailed units is below and equal to threshold
level T, reneging parameter is denoted by u0. Unit reneges exponentially with
86
parameter vi when all permanent repairmen and j (l '.S: j '.S: r) additional
repairmen are busy. * The additional removable repairmen ·will be available for repair depending
upon the number of failed units present in the system according to a prescribed
scheme as stated below :
•
•
When there are n<T failed units, only C permanent repairmen are
available for repair them.
In case of jT '.S: n < (j + l)T, i = 1, 2, ... ,r-1, there are i additional repairmen
availa1le to provide repair with rate ~ti. The / 1' additional repairmen is
again removed \Vhen queue length drops to iT-L i=l,2, ... ,r. • In case of rT s n <1'1 + S + Y-m failed units, all ( C + r 1' repairmen i.e. all
the permanent and additional repairmen "\Yill be busy in the system. We develop the mathmatical model by taking suitable notations which
are given below : M : The number of operating units in the machining system.
C The number of permanent repairmen r The number of additional removable repairmen
s y
A
a
f3
f3 j
u,ui
~l
~l f
~l i
l"(n ),µ(n)
n
The number of warm standby units The number of cold standby units
Failure rate of operating units
Failure rate of a warm standby
Joining probability of a
standby units are available.
units in the queue when some
joining probability of a failed units w·hen all standbys are
exhausted and j U=O,l...,r) additional repairmen are turn on. Reneging parameters of failed units whin a few standbys are aviable, and no standby is avialable and j (j=O,L.,r) additional repairmen are turn on.
Repair rate of permanent repairmen.
Faster repair rate of permanent repairman when all standbys are
exhausted. Repair rate of ith (i= 1,2, ... ,r) additional removable repairman.
State dependent failure rate, repair rate of units when there are n
failed units present in the system. The number of failed units in the system ;,vaiting for their repair including those failed units which are being repaired.
Pn
87
Probability that there n failed units present in the system in steady state.
P 0 : Probability that is no failed unit in the system. 3 FORMULATION OF THE PROBLEM. We assume two cases for
analysis purpose, which are given as follows :
Case I: C::;Y
In this ca...~ the failure rates and repair rates of the units are state dependent and are given by
}=
and
=
Mi.p+Sa,
+(S+Y -nkx
-:-S-"-Y-
nµ,
Cµ+(n-C)v,
Cµr +(n-C)v0 ,
j I --·
Cµr + ""'µ- -'-(n-C + ].\ -1 L r', lJJ'
i=l
Cµr + f.u,-:-(n-C+rlr, ---
Osn<C
Csn<Y
Y::;n<Y+S
Y+Ssn<T
jT s n < (j + l)'r,j = 1,2, ... ,r
rT s n < M + S + Y - m
O<nsC
C<nsY+S Y+S+lsnsT
... (1)
jT < n s U + l)'r,j = 1,2, ... ,r-1
rT < n s lvf + S + Y - m
... (2)
Using appropriate state dependent rates given in (1) and (2) we can write the governing steady state equations as :
_,_ Sct_}p0 + µp1 = O ... (3)
-(Mi,_+ Sa+ nµ)P" +(Ml.+ Sa)p,.,_1 + (n + l)µp 11 +1 = 0, O<n<C ... (4)
-(1'1!.p +Sa+ Cµ)pc +(Ml.+ Sa )Pc-i + (C11 + V)Pc+i = 0 ... (5)
- [M) .. p +Sa+ Cµ + (n - C)v ]Pn + (M!,p +Sa )Pn-I + (Cµ + (n + 1- C)v JPn+I = 0,
C < n.::; Y ... (6)
-[Mi.p+(S-'-Y-n):x+Cµ+(n-C)v]Pn +[M!.p+(S+Y +l-n)aJPn-1
+ [Cµ + (n + 1-C)v JPn+l = 0, Y <n<Y +S ... (7)
88
- [Ml.,13 0 + Cµ + (Y + S -C)v }Py "'"s +(Ml.I)+ o: )Py _s-1
+lC1--tr +(Y +S+l-C)voJPY+S+l =0
- [Mi.0 0 + (S + Y - n )a + C1--t r + (n - C }p n + [Zvfl.13 0 + (S -"- Y -"- 1 - n )a -1
+ lC1--t 1 +(n+1-C)v0 Wn~l = 0. Y+S<n<T
- [(M + S + Y -T)l,13 1 +Cpr + (T -C }PT+ [Ml.13 0 + (S + Y -'--1-T)a}Pr-l
+lCµr +µ1 +(T-C)v1JPr+1 =0 -
r - '
_ [ (M + 5' + Y - jT)i.13: + Cµr -'- Y J--L -r- ( jT-) ; ~ ~ ·.'"'
; l=l
--;
[(M + S + Y + 1- jT)!cp J-l]PJT-1 + Cµr + ! µi __,. i;ol
-;-1- c + j ~, i IPjT ~I = 0 - '
i= 1,2, ... ,r-1 r j J ~ --
-1 (M + S + Y - n )1.13 _; + C1--t f +I J--l; + (n - C + j )v_,: L 1=1
j ( -\ l C1--tr+::=J--Li+ n+l-C+jpJ JPn-l =0
L i j <f.f+
- r-1
_ (M +S+Y-rT)l.I\ +C1--tr + Ll--t; +(rT+l-C+r}:,-•,_1 PrT i=l
I
+S+Y+l-
·=1,2, ... ,r-1
- r r --- l +((M +S + Y +1-rT);q\_1}P,.y_1 +J µf + l>i +(rT+l-C +r~,.P,-r~i =0 I
L i=l J
r r • --
... (8)
... {9}
... (10)
.. .(11
Pn-1 -t-
... (13)
-l (M + S + Y - n )1.,13 r + C1--t r + ti_µ; + (n - C + r lpn-'-- + s + y -'-- 1- n )i.13r1Pn-l J
:,- - -- l + Cµr -r- Lµi +(n+l-C+r~,. lpn-'-1 =0, rT<n<M+S+Y-m
i=1 J ... (
r ,- l -lCJ--tr + Lµi +(M +S+Y-m-C+r~,. Pm+S+Y-m +[(m+l)l.f3,.JPu~S-'-Y-m-l =0 __ (15) l=l -
89
The steady state solution of equations {3)-(15) by using the product type solution
is obtained as follows :
pn =
where
+Sa
u
1 -Po, n~
O<nsC
()\.fi.[3 ~Su r-c Z'vf/,, So. ' 1 C'.Po. C<nsY
n
f1[C~t-'- -C)v] ,Ll
n
+(S+Y +1-i :p 1 (M.i.f3 + Sa)Y-c S1 .Po, Y<nsY+s
+ (k-C)v]
Y-S .o..(S ... Y -'-1-i}::t.] il[l~fl.l3+(S+ Y +1-i)a]
i=Y-:-s+l i y 1 y s IJ(Cµ+(k-C)v] S2.Po,
Y+S<nsT
+ k=Y.,,.S-1 k=C-'-i
Ti.
f1CM-S-'-Y +l-iX1.t-JT$)'i-JT i=T-1
''ff r Cµ f + t µi + (K - c, j p )I fi 11rr r Cµ f + tYi + (K - c + 1~1 ll k=c~l z=l Ll=lk=lT+lL t=l j_J
T 1 j-1 \
TI[Mi.(30 +(S+Y +1-i)aj f1(i.13if I · xi=Y~S·l T \i=l iS3Po,JT<nS(.j-"-l)T,lsj<r
TI[Cµr +(k-C)v0 ]
k=Y-S-l r. r-l
-"-S+Y-'-1- TI(i.f\ )T i=T~l i=l
n r
TI [Cµf + l:µi +(K -C + r k=rT+l i=l
rT f r , l ]. f1 j Cµr +Iµ, +(K -C+r~,. I k=JT7lL r=l J
x, 1 I ,. 1 (1+1 rr r ; i fl TI 1· C . , ~ S .i Po rT li=1k=IT+11 µr+,Lµi+(K-C+l\ j'I .· , <nsM+S+Y-m
~ ~I P1 J
S. = Ml.~Sa l
µ
1
C!' S2 = (Ml .. (3 + Sa r-c S1
... (16)
90
T y,s
[1[M!,(3+(S+Y +1-i)a] q _ i=Y+l S '-,3 - Y-S 2,
f1[C~t + (k-C)v]
[1[Mi_(3 0 +(S+Y +1-i)a] S _ i=Y-S+l S
4 - T 3
[I[C~tr+(k-C ] k=l:--S-1
k=C+l
Jf-S+Y-m
Now we determine P0 , using the normalization condition LP,,= 1. Now we get r.-=O
c (M' S ),, 1 (·~1' S ·)c 1 ,. (~1' A · S ·)n-C p-i =' A+ a _ _._ 11 A+ a . "\" p Af-! +. a () ~ /l I . (" C' ,t_,, it
n=O µ n, µ 'K=C+l f1fC. i' µ-:- ~R -
.k=C...-:
n
+ s (M' s v , Y+S IT l • A + a )1 -C ) ~ ,,__, i=Y ,1
+(S-"-Y+l-ih]
!l
i:..-.S
'(l[Mi,13 + (S + Y +1-
+ S2 i=Y-1 YTS
n=Y-,-l [1 [C~lr-'- (k-C)u] ll[Cµ+
k=C+l k=C,1
T
I
n T [1[MA-(30 +(S + Y +I-i)a] TI
i=Y +S+l ..,.. Sc i=Y +S-c n ~
...,.{S-'-Y .c;_l-l
n=Y +S+l [1[C~tr +(k-C)vo] k=Y +S+l k·=2:~ -S-I
r:
1 r-1 U+lfT f1(M +S + y +1-iX!,)1'-JT(0J''-JT =---------------' )' z=T-1 I J[1-l (l[1+1 )T Ir C ~ (1k -C - l t-'111= iT-1 '---"--=-[1n-----,i~c--f--{--,,-_-_-_.-l ~l~
l' µf+L.Yi+, - -,-lµ1 - , , ·. µr+.:..-µi+,k-L-+J,V l=lh=IT+l\ 1=1 tc=jT-1 i=l
(fi(A-l3if +.S \i=l
4 rT [ r
k=g:+l Cµr + tµi + (k-C + r )vr
1
(l"-lYT ( l n n . C,ur-'- L k=IT -1 1, i=l
-C+l)vl
!tf-S-Y
I n=rT-~
n (lvt -t- S + Y + 1- iKXp" y-rr r;) ~
*=:: _,
,r-1 n . Cu.· ~ "\' u "7" lk - C + r k: , • ,: L._,• 1 ' T
k=rT-1 i=l
Case-II : C> Y
!.{n) =
and
In this case the failure rate and repair rate are as follows:
M!.+Sa,
·Mi.+ (S + Y - n )a Ml.(3+ (S + Y -n)a,
~i\fi.p(} -'- (s - y - nki., +S..,..Y-n
-:-S-1 -n
nµ
Cµ+(n-C)v,
cµ,. +(n-C)v0 ,
O:::::n<Y
Y:::::n<C
C:::::n<Y+S
Y..i..S:::=:n<T
jT :::=: n < (J + lfr,j = 1,2, ... ,r
rT ::::: n < M + S + Y - m
0< n:::::C
C<n:::::Y+S
Y+S+l:::::n:::::T
=<cu.+ ~u, +(n-C+ 1·L, · / L· ~ ~ P J
jT < n :::=: (j + lfr,j = 1,2, ... ,r-1
Using
i=l r .. ,
C.ur "'""Lµi _,.(n-C+r~r' ==1
rT < n ::::: M + S + Y - m
and we can VicTite the governing steady state equations as :
91
... (17)
... (18)
... (19)
-(M!dSa)P0 +~tP1 =0 ... (20)
-[Mi.+ Sa+ n~L}P11 +(Ml.+ Sa )Pn-l + (n + l)µPn+l = 0, 0 < n::::; Y ... (21)
-[M!.+(S+Y-n)a+nµ}Pn +[lvfa+(S+Y +1-n)a}Pn-l +(n+l)Pn+l =0, ... (22)
-[M:;q3+(S+Y-C)a+C,u}Pc +[M;;_+(S+Y +1-C)a}Pc_1 +(Cµ+v)Pc+i =0, ... (23)
-'-(S+Y-n)a+C~t.;..(n-C)v}Pn +[M!.~+(S+Y +l-n)a}Pn_1
+[Cµ+(n+l-C)v}Pn+i=O, C<n<Y+S ... (24)
-[M7tj30 + Cµ+ (Y + S-C)v}PY+S + (M7q3+ a.)PY+S-l + lCµr + (Y + S +1-C)v0 JPY+S+l = 0
... (25)
92
-lMA.[30 + (S + Y -n}a + Cµr + (n-C)v0 jP,, + [Ml.[30 + (S-r- Y + l-n~]P,,_ 1
+ lC\t1 -r- (n + l-C}u·0 JP11 .. 1 = 0, y + S < n < T ... (26}
-l(M + S + Y -T)/,f31 + C~tr + (T-C)v0 jPT -r-(Mi.[30 +(S+Y+1-T)a]PT-i
+lC~tr +i.l1 +(T-C)v1 jPT+I =0 ... (27J
-l(M + S + Y - jT)?,pi + Cµr f µ, + (JT-C + j-1k_1 + ((M + S + Y + 1- jT)l.13.1-J L 1=1
I i , __ ,
PJT-1 +I Cµr + Iµi +!jT +1-C + j~1i L !=l
= 0 j= L2, ... ,r-1 ... (28)
r . I J -J (M +S + Y -n)l.f3i +Cµr +Iµ;+
-,
-C + Jhj !Pn 7 +S+Y-d- ]P,.,_l L i=l
+ J Cµr + f µ, + (n+ 1-C + j~ij;Pn+I = 0, L i=l
iT<n<(j+ l)Tj= 1,2, .. .,r-l ... (29)
-[(M + S + Y -rT)A.13,. + Cµr + ~µi +(rT +l-C 7 rt: .. _1 ~P-r
.. + [(M + S + Y + l-rT)!..13,._ 1 ~-T-1 + Cµr """J µ,-"- .,.. 1- C + r I:·_ P,7 _ 1 = 0
I • - - ... (30) .:=:
r .r , 1
- (M +S+Y-n)A.13,. +Cµ1 + Iµi +\n-C+r~r !Pn +[(M +S+Y +1-n)l.prJPn-l i;I J
+l Cµr + fµ; +(n+1-·c+;~,. '11P,,__1 =0, L i=I _
rT < n < };If+ S - Y - m ... (31)
-[Cµr + i>i +(M +S+Y-m-C+r}',.lpM-S+Y-m +[(m+l)l.13r}PM-S-Y-n:-1 =0 ... (32} !=l j
The steady state solution of above equations by using the product type solution is obtained as follows :
93
11.Jir· ·Sa.}" 1 n ~.i¥1l_-;- -r0
,
n'. O<nsY
fl [lvf/. _,_ (S _,.. Y + l-ih] [Mi. -:-SajY Po, i y l " \ y1
fl k Iµ" Y<nsC
.k=YTl l
n C n [Mi.13 + (s + y-,- l - (h J n rzyn.-'- (s + y + 1- i)a 1 "-::-~._...-::-.._ i=;{-1
nc k l c S5Po, C < n < y . S Iµ - T
-C~:) k=Y +l j
n Y-S fl [1vf/.p 0 + ( S -"- Y ..,. 1 - i )a] fl [M!.13 + (S + Y + 1 - i )a)
i=Y-S-1 i=C+l S R Y + S < n s T Y-S 6 Q,
] fl(Cp+(k-C)v) -C p_ =
k=C-1
_;y TI (1.p; f J{p j r-jT fI [Micl3o + (S + y + 1- ip] ) i=Y +S+l
k=Y-S-l
n +5..,.y -.,-l-. - ,....,,.., .., l=d -1 i=l
. IT C~tr+fµ;+(k-C+l~jl (I(cµr+(k-C)v 0 ) k= ;T+l 1=1 _J k=l' +S+l
1 x_ S-R
1 f{,uf{_(c~t.r+fµ;+(k-C+lhJ] 1 °' Ll.=1R=IT-l\ r=l )
jT < n s (j + 1 )T ,1 s j < r
n : r-1 \ . n (1"\.1 + S + y - i) n (!cp i f l(i.13,. r-rT 1 f=T-1 . i=l
Cµe + f µ- +(k-C+r\., l J . .t:....J i l ~ Pr ~l ~
n Cu:+-.;;:-" ., ' _L_,~_; -C-:- r ~'r rT
11 ,t::.=rT-1 l=..:. k=JT--:-i
1 ~ n x .. .Ssrr;. t r-1 U-l)T } --., ...., ·1
••
1 n n cµ. + y U; + ik -c..,. i,t.;1 i ........, • ~ "\ / ....
rT < n s M + S + Y - m
i=lk=lT-l\ i=l
... (33)
where
c fI[M!,+(S+Y +1-i}x]
S- = (Mi.+Saf " ' Yl
S = 1=Y+l S 6 ( c \ 5
lJ1~k) jµc
94
Y+S
IT(MA.13+ (S + Y + l-i)a] T
IT[l~D-~ 0 +(s+Y+1-i)a] S -~C+I S
7 - Y+S 6,
IT(Cµ+(k-C)v)
S _ i=Y+S+l S s- T ·7
IT (cµr + (k- C)vo] k=C+l i:=Y TS-1
Now we determine P0 using the normalization condition 1,f~S+Y-rn.
· ) Ph = 1. Then, we get ~ _,. n-={l
p-1 = f (Ml.+ Sar 1 . (.ND_+ Saly c Il[Mi. + (S + y + 1-{}L 0 .i.....J -~ • ' . •• • p
n=D µn n! . yi L.. -'-'=;:.,.!-'--~· --:--....,.------ n:=Y...;...l
IT \~k=Y-1
C n
IT[MA.+(S+Y +1-i)a] Y+S IT[M!.l)+(S+Y +1-+ 85 i=Y +l ( C \ L _,__i=C"-'+-"--l _n _____ _
Lr!~ jµc n=C+i JJ}Cµ+(k-C)v)
n:
Y+S [M!.~+(S+Y +1-i)a] T
+S6f1 Y+S I i=C•l f1(C~L+(k-C)v) n=Y-S-1
n :i -IS-'- Y -1-z £=Y-S-l
-.- ] k=C+l k=Y-S-l
T { j-l
f1(M!.~ 0 +(S+Y +I-i)a~ fl(i.~i ; .. l . 1 S i=Y +S+l \. 1=- J . _
+
1
fr[c.LLr+(k-c)vo] f_._ffuf[_ cµr+±µi+(k-c+1~·1··1J1 k=Y +S+l i l=l e=ll +l 1=1 J
I!
IT(M S Y 1 -x, )n-JTfn "1-JT r-l{j+l)T +++-LA \Pjf
x I I i=T+l I n . J '
.i=In=jT ' n ; C~Lr + I~l, + (k-c+ j~J k=JT+1l i=l
( r-1
I f1(;.,p;f ' s \. !=l
..,- , s rT l r --' l f1 JCpr+I.~1,-r -C+r_pri
k=jT ..... il , i=i j
1 r-1 \I~HT
nTI r - ±µi + [k -C-'- l l i'1 Z=l l:=fT -1 :=l
.\1-S-Y-m
' L:-i?.=rT -l
95
ll
TI (M + S + Y + 1-iX!>pJ'-rT i=T+l
n r-1 ,
TI C~tr + L)ti + (k-C +r~,- (34)
f:=rT -1 i=l
4. Some Performance Measures. After obtaining queue size distribution in previous section, now we obtain some system characteristics as follows : * The expected number of failed units in the system
*
*
*
*
*
*
*
},f-S-':-
E(n)= ") n.P,~ ,__ n=:
Expected number of operating units in the system
.~.r-:--8-:-;e:
E(O)= ) -Y.,.S)P . n ,:o:.=Y-S-1
Expected number of unused cold spare units in the system
=) .:....,., -n ;:=fl
Expected number of u_nused warm spare units in the system
s 1·-s
E(lIYS)=YLPn-"- I(Y +S-n)P,, !1=0 ,"=Y-1
Expected number of idle permanent repairmen
_, -~
;:=(1
Expected number of busy permanent repairmen
E(B)=C-E(I).
Probability of /h (1:::; j:::; r -1) additional repairman b~ing busy
, r-1 (i~l )T
E(Aj )=Ii "IP,, .J=l n=jT-1
Expected number of busy additional removable repairmen
r-i fj-1 rT J!-S-+Y-rn.
E(BAS)= Li IP11 +r IP,, j=1 n=jT-l r:.=rT-l
.. .(35)
... (36)
... (37)
... (38)
... (39)
. .. (40)
... (41)
... (42)
96
5. Special Cases Case I : Model with Cold Spares, Balking Reneging and Additional Repairmen. If S = 0 then our model reduces to model \vi th cold spares, balking,
reneging and having additional repairmen.
(a) For the case C s Y ,
p = n
where
), p=-,
µ
we determine steady state probability distribution as
/n!Po, O<nsC
( ~r R)n-C Lvl.Af--' A.Po, C<nsY
fl(Cµ + (k-C)v] k=C-'-l
(1\11,130 r-Y B n .ro,
fr[cµt +(k-C)v0 ]
Y<nsT
k=Y-l n 1 )-1 '\. .
n( ~,.r y l -~IT(' R ·}T '(· f:! \n-1T 1¥1. + + - q 1'-1-'i i 1·1-' j J ,
i=T+l \i=l J 1
Cu,-+)u+ -C+l~-• 1 ~~ l [' l
i=l
n [ J -k=g+i Cµr + ~µi +(k-C+ j~J l j-1 (l..cl)T
IIT IT , l=lk=l'T~l
< n s j_.;._lfT.lsj < r
IJ(M + y + 1-i{ ITU-13i i=T+l \i=l
l ·,-:DPo, ,,.. , ; _; rT ~ 1 --·, j) -" , - - -'- '\' u -'- lk -C ~ 1 b1 i I ,:1 · ' j ) J
n r-l . .
IT Cµr + Iµi +(k-C+r~, k=rT+l i=l
nn k=."'T-l
rT<nsM+Y-m
A= (Mpf. Cl
' .. ,,·Y-C
B = (M1.p} A y ,
f1[Cµ + (k-C)v] k=C-1
c C=
(Mf-Po)T-Y T B, [J[Cµf +(k-C)v0 ]
D= rT r
TI Cµf +Iµ,-'- -C-"-r'b_ ... (43)
k=Y+l k=jT+l i=l
Case-II : Model with Balking, Reneging and Mixed Spares. By setting r=O
our model reduces to machining system with mixed spares, balking and reneging.
p = n
Now for case C> Y, we get steady state probability distribution as
!; 1 +Sa -Po,
n! µ n
(Mi., Sa)·Y . n [Mi,+ (S-'- Y + 1- i)nJ ______ :..!.____ '=} + l 1
Y! n
I1 .~=Y-l
C n
Po,
TI[M!. + (S-'- Y -"-1-i)et}c CT[M!.13+ (s + Y + 1-i)a]
0<n5Y
Y<n5C
•=Y-1 . k=C+} S-P..C<n5Y+S C \ i ll "\ 0 0'
IlkJµc I I1(Cµ+(k-C)v)j k=Y~2 \k=C+l )
97
' , /l
-c-Y~l-C+iµj I1[M1"13o+(S+Y+l-i)a] \l[,w.0" ,,. S·l s,.P,, y + s <no; T ' c-1 TI [cµ r + (k - C)vo]
k=Y+S+l
... (44)
Case III. If 13 = u = 0 then our model reduces to Moses (2005) model for machine
repair problem with mixed spares and additional repairman.
Case Iv. For Y = 0, S = 0, 13 = 0 v = 0, we get results for classical machine repair
problem discussed by Kleinrock (1985).
6. Cost Function Our main aim in this section is to provide a cost function, which can be minimized. to determine the optimal number of repairmen and spares. The average total cost is given by
Y-S r
E(C)=CM L~fP" -'-C1E(I),CscE(UCSh·CswE(UYS)+C3 E(B)+ ICAjE(AJ(45) n=O j=l
where
CM = Cost per unit time of an operating unit when system works is normal mode.
C1 =Cost per unit time per idle permanent repairmen.
C sc = Cost per unit time for providing a cold spare unit Csw = Cost for unit time for providing a warm spare unit CB = Cost per unit time per permanent repairman when he is busy in providing
repair. C:i.i = Cost per unit time of jth(j= 1,2,. . .,r) additional repa:irman.
98
7. Discussion. In this study, we have developed machine repair model \vith
balking reneging, spares and additional repairmen. The machining system
considered consists of warm and cold standby spares along \vith a repair facility
having both permanent and additional repairmen. The provision of spares and
additional repairmen may help the system organizer in providing regular magnitude
of production up to a desired grade of demand in particular when number of failed
units is large. The expressions for several system characteristics and cost function
are derived explicitly which can be further employed to find out the optimal
combination of spares and repairmen and might be helpful for system designer to
determine appropriate system descriptors at optimum cost subject to availability
constriant.
[lJ
REFERENCES M.O. Abou El Ata, State dependent queue: 31 },f l S •,•,·ith
Microelectror Reliab., 31.No. 5, 1001-1007. eue"-m"- ru"1d general balk functions,
[21 M.O Abou Ei Ata and A . .t\LA. Hai-iri, M_\fCX queue with bai.king and reneging, Comp. Oper:
Res., 19 No. 8 (1992), 713-716.
[3] R.O. Al-Seedy, Queueing with fixed and variable cha.Tu"1els considering balking and renegaing concepts, Appl. Afoth. Comp., 156, No.3-15, (2004J, 755-761.
[4] D. Gross, Khan, M.D. and J.D. Marsh, Queueing model for spares provisioning, Nau. Res. Log. Quart., 24 (1977), 521-536.
[5] S.M. Gupta, Machine interference problem with warm spares. server vacations and exh:ausri,•·e serivice, Perf Eval., 23 No. 3, 195-211.
[6]
[7]
M. Jain, M!M!R machine repair problem v.ith spare.sand and:tior"i Appl. Math., 29 No. (5), (1998), 517-524.
lndian J. P,.£:re
M. Jaln and Premlata, Mi MIR machine repair Sci., 132 (1994), 139-143.
w]fo and spares, J. Engg. Appl.
[8] M .. lain, and P Singh, M!Mim queue with bai.king. renegi::g and addi:cional servers, Int. J. Eng, 15, No.112002), 169-17&.
[9] M. Jain, G.C. Sharma and R.R. Pundir, Loss a.'1d delay multi server queueing model with discouragement and additional servers, J. Raj. Acad. Phy. Sci., 4, No.2 {2005), 115-120.
f 10] M.Jain, G.C. Sharma and .M. Singh, MJlvfiR machine interference model with balking, reneg'illg, spares and two modes of failure, Opsearch, 40 (2003), 24-41:
[ 11] M. Jain, M. Singh and K.PS. Baghael, lvf/:.\1,'C!K/N machlne repair problem v.ith balking, reneging, spares and additional repairman, J. GSR, 25-26, 49-60.
[12] J.C. Ke, K.H. Wang, Cost analysis of the 1'.1ilvf!R machine repair problem with balking, reneging and server breakdown, Oper. Res., 50 (1999), 275-282.
[ 13] AI. Shawky, The single server machlne interferences model with balking reneging and an additional server for longer queues, Microelectron. Reliab., 37 (1997), 355-357.
[15] A.I. Shawky, The machine interferece model l;.1/MiC!KiN with balking, reneg'illg and spares, Op .earch, 37 No.1 (2000), 25-35.
Jnanabha, 37, 2007 rDedicated to Honour Professor S.P Singh on his 701
h Birthday)
A NOTE ON MHD RAYLEIGH FLOW OF A FLUID OF EQUAL KINE~iATIC VISCOSITY .. ~'ID 1\'iAGNETIC VISCOSITY PAST A
PERFECTLY C01'i'TIUCTING POROUS PLATE By
P.K. Mittal, Mukesh Bijalwan Department of Mathematics Government Postgraduate College,
Rishikesh, (Dehradun)-249201, Uttaranchal, India
and Vijay Dhasmana
99
Department of Mathematics, KG.K College, Moradabad, Uttar Pradesh, India
iReceir.;ed : August 4, 2007)
ABSTRACT paper deals the of a viscous incompressible fluid of small
electrically conductivity past an ir.Jinite porous plate started impulsively from
a constant transverse magnetic field in fixed relation to the
of small uniform suction or injection velocity at the
plate. Suction or injection velocity at the plate has been calculated using the Laplace
transform method. The MHD unidirectional flow of a viscous incompressible
of small electrical conductivity near an infinite flat plate started impulsively
from the rest, which was first studied by Lord Rayleigh [8], has been shown to be
self-superposable and an irrotational flow on which it is superposable is
determined. Some observations have been made about the vorticity and stream
functions by using the properties of superposability and self-
superposability. profiles have been plotted and studied for different
conditions and for
2000 Mathematics Subject Classification: Primary 76A10; Secondary 76B47.
Keywords : MHD Rayleigh flo-...v, KinematicViscosity, Magnetic Viscocity
1. Introduction. The folw about an infinite flat plate which executes linear
harmonic oscillation parallel to itself was studied by Stokes [ 11] and Rayleigh [8].
The impulsive motion of an inffinite flat plate in a viscous incompressible magnetic
fluid in the presence of an external magnetic field was studied by. Rossow [9]. Nath
studied the Rayleigh problem in slip flow with transverse magnetic field. In
the present paper we have obtained the exact solution for the MHD flow of a fluid
of equal kinematic viscosity and magnetic viscocity past a perfectly conducting
porous plate by using the Laplace transform technique. Vorticity and self
superposability of the above MHD Rayleigh flow have also been studied.
100
2. Formulation of the problem. Let us consider the unsteady motion of
a semi-infinite mass of incompressible, viscous, perfectly conducting fluid past an infinite plate. Let x-axis be along the plate parallel to the flow direction, y-axis
perpendicular to the plate and z-a..xis perpendicular to both the x-axis and y-axis.
The imposed magnetic field H 0 is applied in the direction of y-a..xis. Let v. represents
the suction velocity at the plate, then by equation of the continuity
~/'. = 0 c..,
Also the condition, that at y=O, v=v5
leads to every \vhere. Due to motion a magnetic
field H x is introduced in the flow direction and from the symmetry of the problem
all physical variables will be functions of y and time t onl:y: Lee the pfate be started
impulsively from rest with a constant velocity U, and subject to the conditions :
at t=O u=H =0 y>O ' r '
at y=O u= U. H =0 t>O ' ' x '
as y--.+oo,u=O, Hr=O. . .. (2.1)
The differential equations governing the fluid motion are given by [13]
- - -.2 < . 1 <U1 f'Ul (' U1 - -+v.-- =a--=u--
"'t ~ - - - 2 c ry (y ry
"."\ - -. -2 C'U2 CU2 CU2 C U9 --+v --=-a--+u----s -. "."\ ...... 2 cy CY CV
~ ·~ ct
where
U1 =u+~H x
u2 =u-~Hx
u is the x-component of fluid velocity.
A.= Mag~.etic viscoc-ity of the fluid, a= .jµH0 / p
ii= Coefficient of the viscosity of the fluid, ~ = ,jµ/p
µ = Permeability of the medium.
Applying Laplace Transform to equations (2.2) and (2.3) we get,
d 2u1 (a -vs} du1 pu1 --+ =--dy2 v dy v
and
... (2.3)
... (2.4)
... (2.5)
... (2.6)
101
{a - vs) du2 pu2 - --==--, ... (2.7) v dy v
where u1,U:2 are the Laplace transform of u 1 and u 2 respectively and p is the
kernel of the Laplace transform. The solution of equation (2.6) and (2. 7) are given by
2u 2 i: U1 ==
- u. )v . ----'>'--"--•- -- _::..._
a-v. ___ ,_
or
2
+4p L'
B exp _ (a - us )y 2u
I 2 l
y f(a-u 5 J + 4 ~~ 2\i\ u ; uj
· · ·) {. I 2 f 1 la - v. )v • v ' (a -·L' ) - ' a - u. \ 1 == expi - ' ·• · · . _q_ e.:;,.µ .:::... ... •. ' s · + 4 P + B exp _ l_ .!(--· I + 4 P J
, 2L' . 2 ~ .. U V 2 V U ) U (2.8) J
and
- ( (a.-v .)v uz =exp1-. s.. C \ 2u
v (a+ u ·) • _. ___ s
2 \' v + 4 ~ + Dexpt- Y /(a+ us
l 21J\ u
Applying Laplace Transform to initial conditions we get,
at
and as
We have
u - u - 0 ,, > 0 Ul == - 'Uz == p Y- ,L p
y--+ :t:.'ii=O, H, ==0.
A=C=O and B=D=L~ Equation (2.2) and (2.3) then become
u1
= U exp~- (a-vJy _ Y /: (a-vJ'2
+ 4 p p i 2v 2 V\ u· v
l
[),,. 11(. ) lr( )12 1 _ _ a - u5 y y 1 a - U8 1 ,
4 p :.·
u 2 == -exp1 -- 1 I ..,.. - r P I 2u 2 \! v / v 1
p l + 4- ~(2.9)
u I I
... (2.10)
... (2.11)
Taking inverse Laplace Transform [7] of equations (2.10) and (2.11) and
102
then substituting the values of u 1 and u 2 in (2.4) and (2.5) finally ;,ve get
U . f y +(ex - uJt' . . y-(Ci. + 1,·,)t u = - erfcl ~ erfc· .
2 !, 2-Jut 2-.Jut . .. (2.12)
and
U Jp ( v -"- (a - u )t , ' v - ( o:-:- i· )t H = - I- er'c: - · ' -er1b - ' x ') \! I' , , I' • ,
,.,, v ~t \ 2 ..... ut , ·, 2-v· ut ... (2.13}
flow equation indicates the absence of ex-ponential factor which means that there is no Hartmann layer in the ultimate state.
3.:Flow Superpsoabile on Rayleigh Flow. Let us suppose that a flow
v = (v .... ,vY"uz) ... (3.
is superposable on the flow (2.12). Here vx, 1:z are independent of x and z i.e., these are either functions of y alone or constant.
Applying the conditions of superposability of t\vo fl~ws, laid do•vn by Ballabh
[2] i.e., the two flows with velocity u1 and u2 are mutually superposable to each
other if
curl[u1 x curlV2 + u2 x curlv1 ] = 0
we get,
A vy=(ou!cy)' ... (3.3)
where A is constant.
If a= O, v 5 =0 i.e., when there is no suction or jnjection and magnetic field, we get
uy = -A&t" exp&2 /4ut) ... (3.4}
If we consider the motion in z-x plane only and u2= constant, then from (3.11 we
have
( r--: (?J 'l \ v•= O,-A-vnvtexpy-/4vt. uz}
From (3.5), we readily have
Curlv =0.
This means that the motion denoted by (3.5) is irrotational. Hence we can say that
an irrotational flow denoted by (3.5) is superposable on the flow (2.12) under the
condition cx = 0,u" = 0.
It was shown by Ballabh [3] that an irrotational flow is superposable on a
103
rotational one if and only if the vorticity of the latter is constant along the stream
lines the former. equation of the stream lines of the motion ( 1.12) can be deduced as
x= constant and z=Derf(v) . 13.6)
Hence the vorticity of the Rayleigh flow is constant along the curve (3.6). 4. Self Superposability of the Flow. from equation (2.12) we have
curl[u x curlu] = O. . .. (4.1)
This is in accordance with the condition of self-superposability laid down by Ballabh [3]. Hence the flmv of viscous incompressible fluid of equal kinematics viscosity and magnetic \iscosity past a perfectly conducting porous flat plate, started impulsiively from rest in the presence of transverse magnetic field is selfsuperposable.
It was found Ballabh [3] that, if the axis of the symmetry in the axially symmetrical be x axis and the axis perpendicular to it be R-axis, The condition for self-superposability the flmv \Vill be
... (4.2)
'\1ihere ,.: is of flow,
i our case reduce to
is any function of the stream function rp.
Condition
... (4.3)
Since the a.xis perpendicular to the flow in this case has been taken as the y-axis.
Now if a= O and us=O, equation (4.3) yields for the flow as
. u I= exo
')
y-
~y .. ,,,_ :':L't ·kt . ... (4.4)
It is now evident that for the flow (2.12) under the condition a= O and
us=O, the right hand side of equation is a function of y at any instant. It means that at any particular time the stream function of the flow can be denoted as
'¥ = '¥ (y) . . .. (4.5)
Thus the stream function q; of the flo\v is a function of y and is in the direction of
z axis i.e., in the direction perpendicular to the axis of flow and the direction of the magnetic field both.
5. Vorticity of flow. From equation (2.12) we get the vorticity of flow as
u v+(a-u )t ~ s
,-2.,,Jut
( ( y+(a+vJt12
\llj + expl -
1 2 r:t I
I I -ym; / J ' ... 'j \
... (5.1) \._-. j~ 2-v ;c()f
104
Let the motion is such that u =u and a =2u =2u s . s
then,
( ( u I ·(/y ,11 ~ = ~)expl - -=-,---yvt - 2.J nut 1 2-../ ut 2
t \ '
6. For Injection or Suction.
( i y 3 ,--
+exp!' - ~, +-..Jut 2-vvt 2
\
Case I. When t= 1/v we have
r U ~ ( ( l)2 l . 9 '"" y+ i \v+3)--;- = -- expj +exp1 - -~ ' Ti 2 ,CI ' 4 ' ' lJ -v " i. :, ' 4
Case II. When t=2/u, we have
f ( «) ~ U i v+?'
2 - I 8Xpj - ~ ' - r ..;.. exp
., '-, (y+ 6}~
U 2.Y2n i 8 \_
Case III. When t=3/u, we have
s u r / U =
2 13 ~ exp! _ (y + 3 )
2
.y un L \ 12
Case IV . When t= 4/u , we have
8
') ( ( v+ 9r i~ _- -+expl - 12
\
~- u ( (y+4)2 - l iy ... l2i2
- =---;= exp1 - + exp1 - ----U 4.Jn ~ 16 i 12
... (5.2)
... {6.
... (6.2)
... (6.4)
7. Results and Discussion. To observe the quantitative effects on vorticity field numerical results have been calculated and plotted for above four cases. It is clear from the graph that the vorticity is maximum at the plate and it decreases as we move away from plate. At small times the vorticity near the plate falls abruptly and then it decreases and become steadier as we move far from the plate. As time increases the fall in vorticity becomes less sharp in comparison to that t= Thus as time increases the vorticity tends to become zero throughout. Thus after large tine we may expect an almost irrotational flow.
105
::::> -
0.40
0.35
0.30
0.25 I\
\\
~ 0.20 \ >.. -u -I-. 0 0.15 > t=3/u
~, / / /t=4/u -I ~. •-0.10 --------·--....... ----...____... ~ ,,..,~ ~ . ._______, - ... - ... _ ... ~ ... -r'...:::::e::::::: ::::::: 0.05-t-*-*-*-*-*-*-t-t_ -1
0.00 I t I I I i I ' I 1 I I I I I l I I I I I I I 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
y
Vorticity Profile for Suction or Injection
106
REFERENCES [ 11 M.Ya. Antimirov and E.S. Ligere, Analytical solution of problems of inflow of a conducting fluid
through the lateral side of a plane channel in a strong magnetic field, Magnetohydrodynamics 36.
No.I (2000), 41-53. i2l R. Ballabh, Superposable fluid motions. Proc, Banaras Math. Soc. Ii.VS), IH1940), 69-79.
131 R. Ballabh., On two- dimensional superposable flows, Jour. Indian lvfath. Soc. 16\19421191-197.
141 G. ilm-nes, and KB. MacGregor. On the m~o-newhydrodyna..rrjcs of a conducting fluid betri1een rn·o
flat plates, American Instituie of Physics i 1999 i.
i5J J. Bears, Dynamics of Fluids in Porous Media, . ..\merican E!se\ier Pub .. Co., New York i 1972:.
[6] Chang and Yen, Rayleigh's problem in magnetohydrodynan1ics, Phys Fluids, 2 19591, 393-403. 71 A. Erdelyi, Tables of Integral Transforms Vol-I, _',ic Graw Hill Book Company, New York.
[8! L. Rayleigh, On the motion of solid bodies through YiscDus liquid, Phil. },fog. Series VI, 21(1911),
697-711. [9] V. J. Rossow, On flow of electrically conducting fluids o•:er 2 t1a: in tl-te presence of a transver.::'>€'
magnetic field, Natl. Advisory Comm., Aeronaut. Tech. Sate _\"o.3971•1957 10] M.R. Spiegel, Mathematical Handbook and Tcbi€s. :.k Graw Hill Bo-0k C-0mpany,
New York.
11] V.J. Stokes, Afath., Phys. Papers \Cambridge!, 31.
12] J.K Thakur and D.R. Kiury, Rayleigh flow of a fluid of small kinematics and large magnetic
viscc sity past non-conducting porous plate, The Y<fathmatics Education 20, 111986) 2.3-27.
[ 13] J.K. Thakur and D.R. Kiury, Rayleigh flow of a fluid of equal kinemarics and magnetic
past a perfectly conducting porous plate porous plate, The Mathematics Education 21, 1, (1987
53-58.
I 14J G. Nath, Rayleigh problem in slip flow v.ith transverse magnetic field,
23, 1 (1971), 315-323.
151 A.E. Scheidegger, The Physics of Flou; Through Po TD us L Ton)ntn Press 1100,;3,j_
Jnanabha, 37. 2007 to Honour Professor S.P. Singh on his 701
1; Birthday)
ON SO:ME NEW M1JLTIPLE HYPERGEOMETRIC FUNCTIONS RELATED TO LAlJRICELLA'S FUNCTIONS
By R.C. Singh Chandel and Vandana Gupta
Department of Mathematics D.V Postgraduate College, Orai-285001, Uttar Pradesh, India
E-mail : [email protected]
!Received : October 28, 2006) ABSTRACT
The perpose of the present paper is to introduce some new multiple hypergeornetric functions relmed to Lauricella's functions and intermediate LauriceHa's functions v,ill include as special cases some of functions of three variables due to Lauricella [9] and Saran [10]; and four variables due to
multiple hypergeometric functions of several variables due to Chandel [1], Chandel-Gupta [2], Karlsson [8], confluent
forms due to Chandel-Vish;,vakarma [3] and Vishwakarma [12]. We also introduce their confluent forms. Finally we discuss their special cases and convergent conditons. 2000 Mathematics Subject Classification : Pramary 33C65; Secondary 33C70 Keywords : Multiple Hypergeometric Functions, Lauricellas Multiple Hypergeometric Functions, Confluent Forms.
I. Introduction. Lauricella [9] introduced four multiple hypergeometric
function's · and r:·' •vhich bear his name. The \Yell known confluent
hypergeometric series Lauricella s functions are .1 and . Exton [7]
introduced two more quite applicable confluent forms =.'t and ¢~' 1 of Lauricella's
functions. Exton ([6],[7]) considered the two multiple hypergeometric functions
related to Lauricella's F~' 1 • Prompted by this work Chandel [1] also
introduced and studied the function i~ 1iEg') related to Lauricella's FJ:n). Further,
Chandel and Gupta [2] introduced multiple hypergeometric functions related to Lauricella's functions (later on called Intermediate Lauricella's Functions):
ffrr!nl 1 .~1"' 1 "i W;d,,! and their confluent forms (~<P'.~b, /:i<P~b, /;)<Jit;b, /{)$~b, /ij<Ji~b
Prompted by the above work, Karlsson [8] similarly introduced one more
intermediate Lauricella's function U:JF~;J. Further, Chandel-Vishwakarma ([3],[4])
108
introduced confluent forms t'/' ' i::i~~:~ Vishwakarma [12]
introduced some more confluent forms of above multiple hypergeometric functions:
•l:.J..(n) •h,J.,(n) <k,J.,(11) lk,J.,(n) 15l'f'CD> (6)'f'CD, (2)'f'AD>(3)'f'BD·
Motivated by above wo:rk hare in the present pape1~ we introduce new multiple hypergeometric functions related to Lauricella's functions and Intermeditae Lauricella's functions, which include as special cases some functions of three variables of Lauricella [9} and Saran [10] and four variables due to Exton with muLiple hypergeometric functions of several variables due to Lauricella [9], Exton [6], Chandel [l], Chandel-Gupta [2], Karlsson [8], confluent forms due to Chandel- Vishwakarma [3] and V:ishwakamra [12J. We also introduce their confluent forms. Finally, we discuss their special cases and convergent conditions.
2. New Functions Related to Lauricella's Functions. In this section, we introduce following three new multiple hyper geometric functions related to Lauricella's functions.
( ? 1) (kYlEf"1(. J... b ' " ) ~. . · (1) b a,q,. .. , ,,;c,c ,c ;x1, .. ,x,,
;: ( ' ' Xb ) (b ') .m1 mm ' a,ml T •••• mn. l,lnL···· ,"1,mn. Xi xn L.. ( '. . '' ) ...
,,n, .... ;n,,~o\c,m1 + ... +m1;)(c',mk.,.1 + ... .;_nr.<:Xc",m.,,_1 .;_ ... +m"' ~!
l::;k:s; ,n-c N
It is clear that
(2 1( )) (k,n)E(n)( b b . ' "· . . a Ill D a, i>···· ,,,c,c ,c ,x1 , .• ,x,,
_(kl'E(")( b b . ,'. . ) -(1) D a, I> ... , 12 ,c,c ,xi, . .,xn . tExton's for k'=n)
(2 l(b)) \k,k)Ei"l( J... b . , .~. . . "l . (l) D a,q, ... , n,C,C,C ,.1~1 , .. ,x,,,
_(k)E("l( b b · "· ) -(1) D a, 1, ... , ",c,c ,x1, .. ,x" . (Exton's /NE~ 1 for k'=k)
(2 1( ) . :;. "lEt"l( b b ' " ') . c J · iiJ D a, 1,. .• , n;c,c ,c ;x1, . .,x"
= Ftd(a,b1 , ••. ,bn ;c;x1 , ... ,x") (Lauricella's for k'=k=n)
(2 2) (k,k'lE\"l( , " b b . . . ) . (2) D a,a ,a ' 1, ... , n,C,X1,-·,.Xn
Y:·
I m! .. --.J7tn:::::O
(a,m1 + ... +mkXa',mk+l + ... +mk.Xa",m;-;'-l ....... +m"Xbi>mi) .... (bn,m,J (c,m.1 -'- ... +m,,)
~ ~ . ~ ~ l<k5ok'5on: k,k',nEN.
,··· !' - ' m1' mn.
2.2.
(2.2(b)
=Fg'
{2.3)
a', a'' ,b1 , ... ,b0; c; x 1 , .. ,x,,)
a',~ , ... ,b1,;c:x1 , .• ,x,,),
a ,a" ,b1 , ... ,b:r ;c; .1:1 , .. ,x0
}
,b1 ,. . .,b,,;c:x1 , •. ,x.,)
a'. a" ,b1 , •.• ,b11 ;c;x1 , .. ,.'l::n)
b,,. ,c,x,, ... ,x11 )
' " b ) a·,a , ;c1 , ... ,c,,;x1 , .. ,x"
(E t ' (k)E(n) f k' ' x on s (Zl D , or =n J.
E · ' \h)El"l f k '-k) xton s i·/\ D , or -.-•
(Lauricella's F};d, for k '=k =n).
109
' L., (a,m: -r ... + mkXa' ,mh+1 + ... +mk.Xa" ,mk'+I + ... + m,,Xb,ni1 + ... +m,,)
., . ..... ,_
1n~~ n1~:
(c1, mi ). .. (cn,mn)
l-:;k-:;k'5on; k,k',nEN.
(2.3(a)) , b; c1 , ... ,c,,, :x1 , .. ,xn)
a' ,b;c1 , ... ,c,, ;x1 , .. ,xn). (Chandel's , for k'=n)
(2.3. ( , ,, b ) ,a,a ,a , ;c1 , •• .,c12 ;x1 , •• ,xn
= T
a ~o;c,~·--- . r. l" l • .-~~~-·~-~ J (Chandel's , for k'=k)
(2 al ;x1 , .• ,x.,}
= PJ" 1(a,b;c1 , .. .,c";x1 , .. ,x,.) (Lauricella's F;(n), for k=h'=n)
3. New Intermediate Lauricella's Functions. In this section, we introduce following new intermediate Lauricella's functions :
3.1) !U 1(a,b,b',bk_1,.. :c1, ... ,cn;x1,··,xn)
:::.
I (a,m1 + ... +m 11 Xb,m1 -7- ••• +mkXb',mk+I + ... +mk.Xbk'+I,mk'+I). .. (bn,mn)
m: .. .rr:c:-=D
xr' x:;' ml!··· mn!
(c1 ,m1 ). .• (cn,mJ
1-:;}:r,-:;k'-:;n; h,k',nEN.
110
If is clear that
(k,hlF(nl_(klF{nl (O,k'ivlnl_(k'IF" \TZ.nl'}'\"' _ vtni 11.ll1f''"I -F·1n AC - AC ' r AC - AC' AC - L' C ' AC - A ' == F~~Tl
(3 2) (k,k')F(n)( b b . 1 • • - ) . AJJ a, 1 ,. .. , n,C,C ,C;,-~p---,C_,,,X1 ,..,xn.
f , (a,m1 -:- ... -'-mnXb1'1n1 ): .. (b,,m,J x;'' ... <' m,,.:;;,,=o (c,n11-'- ... + l?l;, Xe' ,mi:-l-'-- ... T l?lio Xck -1,m1:-~1 ) ... (c!1 ,mn) ml: m-,71·
which for special value of k and k ', gives
(3.3)
u .1:1F("l=u'1F1" 1. AD AD·
'Fv:l I «.r: . ~W ~
1(a,a' ,ak'~ 1 , ... ,a 11 ,b1 , .... b71 ;c;x1 , .. ,x")
1 :::; k :::; k':::; n, k,li', n E N .
i_l.l -F(nJ (O,Dl'Fl"J -Fin) - B ' BD - B ''
I n11 ... m"=-0
(a,m1 + ... "'- m.t_Ka' ,mk-l-'-- ... -'- rn;.; .~ak _1 ,mk_i) ... (an,m,.,Xb1 ,m 1i .. (b,,,mJ ( ) \C,nr1 -'- ... +ni"
xnil _l
ml l'
xmn ~
mi' n.
which suggests
(k,k)F(n)_(k)F(n) (O,k')F(n)_(k')F(nJ !n.n B D - BD ' BD - BD '
- (~.!..
(3 4) (k,k iF(n)( b b' b b · 1 r· • - ~- l , . CD a,, , i, ... , k,c,c,c,,,-~i.' ... ,t."'1.:, ...... ,.,_,
"' I (a,m 1 + ... +m 11 Xb,mk-l -:- ... -'-rn;: .m~-- - ... -m" ,,1n,L .. (bk,mk)
1n 1 •... ,nz11
=0 (c,m1 + ... +mkXc',m::-: -'- ... -:-m;, Xc,,_ 1 .m1:_1 } .. (c",mn)
JJi, X1 ,
m1!
so that
xml! n
m I' n·
(k,k)F(n)_(k)F(n) (O,k')F(n)_(k')F{n) (n,niF(l'; - F\nl 'l,lJF(n!_il.IE'·"i CD - CD ' CD - CD ' CD - D ' CD -(1,' ·c '
4. Confluent Forms. In this section, we introduce following confluent forms of above multiple hypergeometric functions :
1. (l,,h'!E(")( b b . I II. - ') (4.1) lffi0
(l) D a, 1 ,. .. , n,C,C ,C ,cx1 , . .,CXk,Xk-l, ... ,Xn C-7
I (a,ml + ... + m,, Xb1 ,ml ) ... (bn ,mn) \ x~' ... xn .
m, ..... m,,=o(c',1nk+l + ... +mk.Xc",mf: . ..,. 1 + ... ~m"J ,n,: nz.-=:
b b ' " ) :, . ., n;c ,c ;xp···,xn,.
Si milady,
(4.2)
{4.4)
, .. ,bn ;c,c";x1 ,. .. ,x11)
= f (a,m1 + ... +mnfb1 ,m 1 ~..(b 11 ,mJ_ xZ'1
...... 1 c . . Xe" . . ) , xm/1
n
m! ml,-·~ .. .t:n4,=l}\ ,mi T ... -t-mf:. ~m-}:;_-~";. ! ..• -rm!Z ml; n·
(a, b1 , •. , bT:: c, c·; x1 , •• .,xn)
O'. ( ' ' x· ,b \ (b ) m1 , mn L a,m1-'-T .. ~'7'mn 'l,ml /.~· n~mn X1 ! ... xn !
,,,,, ... J71.,=o(c,m1 ..... mr.Xc ,mk-1 ..... mk.) m1. mn.
x. x. \ , ,. , b · l n j a.a .a .o. , ... , ,, :c;--, .. ,--,xk+i '···,xn · · a a .C~:J:
111
"' I
m -'- -'- V ,, -'- + m Xb ) (b ) . m, ,. ·<-: .... m;:!\a ,m;,,·~1, ... " l,ml ... n>mn Xl Xm" ll
=f.,- (c,m 1 -'- ... +mn) m 1 l m' n·
(4.5) ' " b b Xk+l Xk' X a,a ,a·, 1,. .. , n;c;x1' .. ,xk,7'"""'7'xk'+l,. .. , n
>:
:L (a,m1 + ... +mkXa",mk'+l + ... +mnXb1,m1 ) ••. (bn,m11 )x~'1
(c,m1 + ... +m") m 1 ! ' .. ,4 m!-- .Jn.~;=0
a" ,b1 , •• ,b"' ;c;xi. , ... ,x")
Similarly
(4 6) !k):'),,,ln)( • b b ) . (2)'+'D.1 a,a' 1, .. , n;c;xl,. .. ,x,,_
O'.
I m 1 •••. J1'i_.,.-=0
( · · X · · · Xb } (b ) m1 a,ml T ... --r-mk a ,mk+l '7' ... +mk' l>ml .. n'mn X1
(c,m1 + ... +mn) m 1!
' ) · \k,k') 1,n)f , " • . • X1 Xk (4.7) Inn (l}Ec I a,a ,a ,b,c1 ,. .• ,cn,-, . .,-,xk+I>··.,xn a- \ a a
m X n n
m' n·
xm!l n
m' n.
(a',mk~1 + ... +mk.Xa",m,1;·+1 + ... +m"Xb,m1 + ... +mn)xri "' = :L
m 1 , ••• ,..mn=0 (cl ,ml ) ... (en ,mn) m1 !
xmll n
m' n·
112
I( ' " b ) , a ,a , ;c1 , ••• ,c,,; x 1 , ••• ,x11
•
1. \k.l'iE\n) 1 " b· . xhl X1: -(4.8) im lll c a,a ,a, ,ci, ... ,cn,x1 , •• ,x1c,--,-,. . .,--,,x,1:._1 , ••• ,x a -;oz a a
I (a,m 1 + ... + mk Xa" ,mk'+I + ... + m" Xb,m 1 + ... + mn) x~''
(c 1 ,m1 ) ... (c11
,mn) m1 ~ 111 1 •... Jn,, ::;:;:{)
(h,1.-'!,t,{n I( ,. b b ) = t li 'f' c" a, a ' i ,. ., " ; c; x l •... , x" , .
(4.9) l~m a -t:J:
, " b . . Xk'+l X,, a,a ,a , ;c1 , ••. ,c,, ,x1 , .. ,xi: ,x!:_1 , ••• ,x,,.,--_-, ... ,
"' y (a,m 1 + ... +m1:Xa',m,1;_ 1 -'- .•• "'"'mi: Xb.m 1 -7- ••. -'-mr.)x~' ( \ ( \ ~ c ! , m 1 ) •.. c n • 1n ... ) m, .
x~i;,
n ... m,,=0 m' n.
{k.k'l,t,(n)( • b ') = (l)'t'c_, a,a, ;c1 ,. .• ,c";x1 ,. .. ,xn
(4.10) · {k,k') (nl ,, . _Xi X1: • l~ FAc a,b,o ,bk'+I, .. .,b,,,c1,-·.,cr:,b' .. 'b'xk_1,···•xk.,xk.+1
I (a,m 1 + ... + m" Xb' .m~-- 7 ... +rni: -:·n';;;-: ,n2,.,)
nz 1 , ..• ,1n11
=0 (cl. ,,,m
_(k,k'),i_(n) ( b' b b ' · ·.. · 'l - (l)'PAC a, ' k'+l'""' n,Cl, ... ,Cn,xl,. .. ,:i.,,,..
(4. 11) 1. (k,k')F(n) b b' b b . . - . Xk+l 1m ·ca,, , ;,,.·_., ... , ,.,c,, .. .,c_,x 1 ,..,xk,--b·-,:1: f1. - ' l d - - - - bl
x,, ~··-, --,JC, b' . ,;+l, .. .,X,,
(4.12)
Y: (a,m1 + ... +lnnXb,m1 + ... -'-mkXbk_1 ,m;;-_1 i .. (b",m,,)x~' (c1, m1 ) ... k,, 'm,,) m1 ! I
nz 1 , ... ,ffi 11 =0
_(k,k'),t,(n) ( b b b . . . ') - (2)'t'AC a, ' 1"+11··1 n1C1,. .. ,Cn,X1,···,Xn ·
lim (".k')F(ni{ b b' b b . . - x.l::'-l x"' ACI a, ' ' k'+l"'"' n>cl, .. .,cn,xl> .. ,xk ,---, .... --
bi..-' -1 , .. /;,/ -+:-.J: \ b. ~ ~. ~ - ...
I n11 , ... ,,rn 11 -=-0
(a,m1 + ... +mkXb,m1 + ... +mkXb',mk-i
(c1 ,mJ .. (cn,mJ
- ... -'- m;; ) x;''
m 1 l
:r;~
fn I n.
xm·., n
ml n·
-
113
(a,b,b' , .. ,bn ;ci , ... ,en ;x1 , ... ,Xn)
Similarly
14.13)
4.
!4.15)
x
) .t......
lirn ;. ;., - .-""'! . -~' --.
!.a,b';c 1 , •.. ,en ;x1 , •.• ,x10)
(·a , . lib' . . ) m, ~ -~m 1 -r- ... 7mr:.l\ ~mr.:_ 1 -r- ••. 7m,1i, x 1
=G (c1 ,m 1 t .. (c,, ,m,,} m1 !
X1 Xn a D' o' :c.c' c,__~ ,. .. ,cn;-b , .. ,-b "- 1 , •••• - , '
1 n
xntll n
ml ,,.
z ( , . ) ml m .. y . .a,m1 -c- ••• -c- mn X 1 ... X 12 "
m.'.":;;', ·=a(c,m, + ... -mhXc',mk_1 + ... +mk-Xck.+1,m1,+1) ... (c,,,m")m11 m 11 t
c, '.:'. ~ ~ ~ :xL , ... ,xn)
: c,c·, C1; _1 , ... ,en.; cx1 , ... ,cxk, xk+l , ... ,xn.)
x (' . , X' ) (b ) m, mn I .a,m1 - ... -r-m"· o1,m1 ... ",m,, l ... ~ m
1 .... .ni,=0(c',mk~l + ... +mk.xck . .,.I,mk'+l). .. (cn,mn) ml! mn!
(a, b1 , ..• ,b11 ; c', Ck·+1 , •.. ,c"; x 1 , ••• , x").
l . u,.k·1Fl"J( b b . , . , . , ) (4.16) im AD a, 1 , ... , ,,,c,c ,ck.+l, ... ,cn,x1, ... ,xk,c xk_1,. .. ,c xk.,xk'+l, ... ,x" c ---:,.'Q
(4.17)
"' (a,m1 -'- ... +m11 Xb1,m1) ... (bn,m,J x;' 1 xmt; n
I .m .. m _ ... Jn,; :;::_lJ ml + -~· - .,_ -i} .. (c,.,,m") m 1! ... m"!
(a, b1 , ... , b"; c,ck'+i •··.,c," ;x1 , ... ,xn).
,.· ( )( XI Xn) • U:,k) n, , .. __ hm FBDI a,a ,ak.-1,. ... ,a.~,b1, ... ,b,,,c, b , ... ,b b; , ... h,->x \ l n
"' I (a,mi + ··· + mk Xa' ,m·t-1 + ··· + mk. Xak.+l ,mk'+l ). .. (a" ,m") x~'
(c,m1 + ... +m") m 1! _____ m,;=0
_(k,k')J,.(n) (. , . . ) - (1)'l'BD a,a ,ak'+I•···,an,c,xl, ... ,xn .
l . lk.k')F(n) • b b . . ~ ~ l (4.18) Im BD a,a,ai:._p····,an, 1,. .. , ",c, , ... , ,xk+l, ... ,xnl a->O: a a )
Xm" n
mt n·
114
" I
(a' ,mh+l + ... + mh. Xak'_1,mk.+l ) ... (an ,m,, Xb1 ,mi}, ... ,(b,, ,mn) x;'' (c,m 1 + ... +mJ m 1 t
111 1 •...• m,,=0
_(k,/;°},i..(n) ( I b b • • . ) - !2J'l'BD a ,ak"+l,. .. ,a", p···, ,,,c,x1, ... ,xn .
1. (k,k')F(n)( , b b . . Xk+l xk. (4.19) :im BD a,a ,ak'+l, .... ,a,,, 1, ... , ,,,c,x1,. .. ,xk,--.-, ... ,--, ,xk'+l,. . .,xn a->~ \ . a a
·r.
I (a,m 1 + ... + m1:Xa1,_1 ,m;;+i ) ... (a,, ,mn Xb1 ,ni1 ), ••• ,(bn ,m,,) x~1:
(c,m:. + ... +m") m 1 ! n? 1 •• .• nz,, =0
l (a,ak'+ 1 , ..• ,a.n ,b1 , ••• ,b,,; c; x 1 , ••• ,x,,).
(4.20) G.J. -l ... G,,->O: lim ' b 7 xk'+l a,a ,ak'+1 , •.. .,an, 1 , ... ,o." ;c;x1 , •.. ,xk,xk+l , ... ,x,,. ,-·-·-
a..1;:+1
, .... ,
-r. ( X I Xb )' {' .) '1'£ •. a,m1 + ... -t-mh a ,mk+l + ... +mk. l,m1 , ... ,\on,m1,, Xi'
xm,, n
mf n:·
x"'· n
ml n·
xn
an
I (c, ml + ... + mn) ml! n11·---.rn"=O ml n.·
_{h.,k')J.(n) I ' b b . . ) - (4)'1'BD\a,a' 1,. .. , n,C,X1,--.,Xn ·
l" (k,k'JF(,,i( b b' b b ' (4.21) im cvla, , , i,. . ., k;c,c '"'k-1···-, a---7~
x, x
a a
I (b,mk+l + ... +mkXb',mk-l + ... 7n1."·Xb1 ,mJ ... ,{bk,mk)x;'1 ••• x~··
m,,. .. ,m,,=0 (c,m1 + ... +m"Xc',mi:< + ... -'-mk'Xc,,+ 1 ,mk.+ 1 ~ .. (cn,mn) m,1 m,,t
_(k,l;'),i..\n) (b b' b b • I f' • • ) - (l)'l'CD ' , 1, ... , i:,c,c ,C,i,·-1,···,~n,X1,.··1Xn .
/
1. {k 1-')F(n)I b b" b b . x,_, X;,,- . (422) i:·1 1
. •• CD a,,·, 1 , ... , k;c,c,c"°-''' .. .,c,,:x1 , ••• ,x,,~.- .. ,-- ,x}._1 , ••• ,x" • b-; \ • ,. • • I • ,. b b • • ,.
f (a,m1 + ... +m:,Xb',mk--d + ... +m,,Xb1 ,m1 ), •.. ,(bk,m,J x~' ... x~·, m1,.-.,m,,=o(c,m1 + ... +mkXc',mk+1 + ... +m.k.Xck.+1,m;,·+1) ... (c",1n") ni1. mJ1!
_(k,k'),t.{n) ( bl b b • J • )' - (2)'1'CD a, ' i,. .. , k,c,c ,ck'+I,. .. ,cn,x1,.··,x,, .
- ( ' 1. (k,k'JF(n)I b b' b b . ' . Xk·-:;_ x,, l (4.23) b~n:. CD[ a, ' ' 1,. .. , k,c,c ,Ck·~1,.-·,C,,,X1,.··,X;,-, b' ,. .. , b'
"°'"' \ j
115
y (a,m 1 -;- ... -'-m,,Xb,mk+1 + ... +mk.Xb1 ,m1 ), ••. ,(bk,mJ x;'' ... x';"
....., =0 (e,m1 + ... +n1e:Xe'.m.,,_ 1 + ... +mk.Xek.+ 1 ,m1,+1 ) ... (en,m,,)m1 l m"!
b,b1 , •.• ,b1;; e,c·, c~-: , ... ,c., ;x1 , ... ,x 11 ).
4.24) Lt.i:. X1 Xk'
a,b,b' ,b1 , •.. ,b.~;c.e' ,c1:_1 , ... ,e,, ;b, ... ,b,xk+1 '··.,xn l k
b-1 • - .. D_.-+:>:
{ ';{ x· ) m m i .'a,m: + .... -mn1\b,m;;_1 + ... +m1,, b',mk+i + ... +m 11 x 11
••• x 12 "
=D (c, m 1 -"- •.. ,,,,_ ni.< Xe', mk-I + ... + mk. Xek'+l' mk'+l } .. (e", m 12 ) m 1 l m 12 !
(a,b,b'; e,e', ck _1 ,. .. ,cn ;x1 ,. .• ,x11).
(4.25) lim a.b,b',b, \
xi . xk xk+l xk. x,, l ;e,e',ck.+l,. .. ,en; b: ,..:'b;'b' ... 'l.j" .. ' b')
') "--'
. . ) "" m1 + ... + mn X1.
le, ml -'- ... "'""m.1: Xe' ,mk+l + ... + mk. Xck.+1,mk.+l ~ .. (en ,mn) ml! ... m" i
yl;~in
~ ( ., ) a;c,e ,e1,_1 , ••. ,c11;x1 , ••. ,xn .
{4.26) 1(a, b, b', b1 , •.. ,b;;; c, c', ck.+I , ... ,c 11 ; ex1 , ... ,exk ,xk+I , ... ,xn)
C--3'>-X
"' L: (a,m1 + ... +mnXb,mk+l + ... +mk.Xb',mk'+l + ... +m,,Xb1 ,m1 ), ••• ,(bk,mk)
m1 -~--..2n.,, =D (e', mk+I + ··· + mk. Xek'+I ,m.k'+l ) ... (en ,mn)
rn. 1 X1
m 1 ! m.,,~
_(k.k'l,t,{nj ( . b b' b b . ' . . ) - 161'+'CD a,' '1, ... , k,e,ek.+1•··.,e,,,x1,···,x,,_.
l . LU')F("i( b b' b b . 1 • I ' ') (4.27) Im CD a, ' ' 1, ... , k,e,e ,ck'+I, ... ,en,x1,. .. ,xk,e xk+I, ... ,c xk.,xk'+I, ... ,xn £'--'>0
L:
x~' l
m1;
(a,1n1 + ... +m11Xb,m1:-1 + ... +mk.Xb',mk'+l + ... +mnXb1,m1),. . .,(bk,mk)
(e,ml + ... +mkXek'+I•mk.+1) ... (cn,mn)
xm~ . " ml " .
116
_!h,k') '(n) ( • b b' b b · · · ) - (7)<JlCD '1, > '1, ••• , k,C,C1;:+11···,Cn,X1,.-·,.Xn ·
1. {k.k'lFlnl( b b' b b · ' · ) (4.28) im CD a, , , 1 ,. . ., k,c,c ,ck.+1, .. .,c 12 ,Xp ... ,x;:,Ck·+iXk·+p··.,cnxn c, .. · 1 ..... c,. -+-0
y:
I m 1 •...• mn=0
xml 1
m1!
Xm" n
m' n.
{c,m1 + ... +mJ:Xc',mk+l + ... +mk.)
_,k,k') (nl( ·b b' b b · · " • - (s1<PcDa,' '1,. . ., J:,c,c,x1,· .. ,x.,).
No doubt all functions studied above are included :i.n generalized multiple hypergeometric function of several variables due to Srivastava and Daoust [11], but they have their own interest.
5. Special Cases. When n=4.
1r..1:1E(nl \1) D
(5.1) C 3iNE~l(a, b1 , b2 ,b3 , b4 ; c,c' ,-; x1 ,x2 ,x3 ,x4 )
= K 11 (a,a, a, a, b1 ,b2 , b3 ,b4 ; c,c, c, c;x1 ,x2 ,x3 ,x4 ) (Exton(?] p. 78, (3Jt 11
,... 2J 12'4 lE 141 (. b b b b ' ·1 D. · (!) D a, l• z, 3, 4;c,c ,-;xi,X2,X3.X4 .'
= K 12 (a, a 2 , a,a, b1 ,b2 ,b3 ,b4 ; c,c,c· ,c' :x: .x2 .x,,, x., 'Exi:on p. 3.3.12.J)
( ,... 3) (z.3 )E(41 ( b b b b · ' "· 1 u. . fi) D a, l> 2> 3, 4 ,C,C ,C ,XpX2 ,x3 .x4 1
=K13 (a,a,a,a,b 1 ,b2 ,b.,,b4 ;c,,c',c";x,,x2 ,x3 ,xJ(Exton [7] p. 78, (3.3.
u.1·i£!11l (2) ,,
(5 4) !3•4 JEC 4l( ' b b b b - · .) . (2) D a,a ,-, l> z, 3, 4,C,XpXz,X3,X4
= K 15 (a,a, a, a' ,b1 , b2 ,b3 ,b4 ;c, c,c,c;x1 ,x2, x 3, x 4 ) {Exton p. 78, (3,3.15))
(5 5) t:.4JE( 41 ( ' b b b b · · ) . :') [) a,a' l• 2• 3• 4,C,X1,Xz,X3,X4
= K 20 (u 11, b:1, b4; b1 ,b2 ,a', a'; c,c, c, c; x 1 , x 2 ,x3 , x4 ) (Exton p. 78,
(5 6) cz.sJEl 4 l( ' " b b b b ) . (2) D a,a ,a' l> z, 3, 4;c;x1,X2,X3,X4
= K 21 (a,a,a' ,a" ;b1 , b2 ,b3 ,b4 ; c, c,c, c; x 1 , x 2 ,x3 ,x4 )(Exton p.78, (3.3.21})
ti:.!.· IE(nl 1il ('
117
5. a',-, b; Ci' Cz' C;1 • C4; x, 'X2 'X;3 'X4)
=Kl b,b,b;a.a,a,a';c,,c2 .c,.c4 ;xpx2 ,x:i,xJ(Exton [7] p.78, 13.3.2)).
- ~) "~'E'~'I ' b ) b.115 ·-··.,·;.··,a.a,-, ;c1,C:2,C:31C4;X1,X2,X:3,X4
= K .s(b,b,b,b; a, a,a' ,a·; c1 , c2 , c3 ,c, ;x1 , x 2 ,x:31 x 4 ) (Exton [7] p. 78, ( 3.3. 5) ) .
. ,.. 9' "''"E'~ ' " b ) (v. J ;,, r. a ,a, :c1 ,c2 ,c3 ,c4 ;x1 ,Xz,X3,X4.
= Kw{b,b,b,b;a,a.a·,a·;c,,c2 ,c3 ,c4 ;xpx2 ,x3 ,x4 )(Exton [7] p.78, l3.3.10))
(,.. 10)' (3.4lp{4J( b b' b" . 'Y ') J). , AC a, ' ' ;c:,C2,C3,C4,X1,Xz,""3,X4
= K., {a,a,a.a:b.b. b,b' :c, ,c2 ,c3 ,c4 ;x 1 ,x2 ,x3 ,x4 ) [Exton [7], p. 78, (3.3.2)]
5.11 ·r· ~ '- -: ~ :x1,Xz,X;3,X4)
=K a.a.a:b.b.b. :c:, ,c,.c4 ;xpx2 ,x:i,x4 )(Exton[7]p.78, (3.3.10)).
12 ' ,Cz,C3,C4;X1,Xz,X3,X4)
=K.~(a,a,a,a;b,b';cI,Cz,C3,C4;X1,X2,X3,X4) (Exton [7] p.78, (3.3.5)).
',.. 13' l:l..ilpt4i( b b b . ' . . . ') \v. 1 AD a, 1 , 2 , 4 ,C,C ,-,x,x2 ,x3 ,x'4
=Ku a, a, a:b,. b2 , b?, b4 ; c, c'; x 1 , x 2 ,x3 , xJ (Exton [7] p. 78, (3.3.11)).
,.. 4) '"-"' 1F"" 1 ' ' ) (D.1 . "' ,04 :C,C ,-:x1 ,X~,X;l,X?.
= K 12 (a,a,a,a;b,, :c,c,c·. c' :x, ,x2 ,x:3 ,x4 )(Exton (7] p. 78,(3.3. l2)).
,.. 1 - "'"·"F'"' ( b b b b ' " .) D.,;,,;:Ji "'a, 1 , 2 , 3 , 4 ;c,c ,c ;x1 ,x2 ,x3 ,x4
= K 13 (a, a, a,a; b1 , b2 ,b:3 ,b4 ; c, c,c', c"; x 1 , x 2 ,x3 , x 4 ) (Exton [7] p. 78, (3.3.13)).
'F~~
(5.16) a',-; b1 , b2 , b3 , b4 ;c; x,x2 ,x3 ,x4 )
=Ki;; {a,a, a, a' ;b1'b:2 ,b3 ,b~; c, c,c,c; x 1 , x 2 ,x3 ,xJ (Exton [7] p. 78, (3.3.15)).
( ,.. 7) (2.4lpl4l( ' b b b b . . . . ) u.l BD a,a ,-, ll 2> 3, 4,C,X,X2,X3,X4
118
= K 20 (a,a,b3 ,b4 ;b1 ,b2 ,a',a';c,c,c,c;x1'x2 ,x3 ,x4 ) Exton [7] p.78,(3.3.20)).
,.. 1s· (2.3lFt4 l( ' " b b b b · · · ) \u. ) BD ,a,a ,a ' 1' 2, 3, 4,C,X,X2,X3,X4
K ( • "b b b b )iE · r~1 ·~s 1 3391··· = 21 a,u,a ,a; 1, 2 . 3, 4;c,c,c,c;x1,x2,x.3,x4. \ xton lf.; p.1 '"' .. - }).
u'-" J F i:;i
9 t2-3JF(4 l( b b' b b · ' · ) (5.1 ) CD a, , , 1 , 2 ,C,C ,C4 ,x,x2 ,X3,X4
= K i:i (a,a,a,a;b 1 ,b2 ,b:3 ,b.., ;c,c,c' ,c 4 ; x 1 ,x2 ,x3 ,x4 ) ,(Exton{7] p. 78,(3.3.13)).
l2 -2lFl4 l( b' b b · · '} (5.20) CD _a,-, , 1' z,C,-,C3,C4,X1,X2,X3,X4
= K 12 (a,a,a,a; bp b2 ,b' ,b' ;c,c, c3 ,c4 ; x 1 ,x2 , x 3 , x 4 ) ,{Exton p.78,(3.3.121).
( .... 2 · t3A 1F 141 ( b b b b · ' · ..... · '1· n. l) 'CD ,a, ,-, ll 2, 3,C,C ,-,X1,·"21X3,X4,
= Ku(a,a,a,a;bpb2 ,b:1 ,b;c,c,c,c';x1 ,x2 ,x:3 ,xJ,(Exton [7] p.78,(3.3.11
( ,.. 22} il.Zl,1,( 4 ) ( b b' b ! ) .D. ' '' ''l'cD a, ' ' l;c,c ,C3,C4;X1,Xz,X3,X4
- K (. · b' b' b b· '· ) rE t - 10 a,a,a,a, ' ' 1' ,C3,C4,C,C ,X3,X.-,,X1,X2 ,, x .on p. 78,(3.3.
( ,.. 23) t2•2 lFl4 l( b' b b · · ) .D. CD a,-, , 1 , 2 ,c,-,C3,C4 ,XpX2 ,X3,X4 ,
- K (a a a a· b' b' b b · r· c " r· · ~ r .,,. E=7"on - 9 ' ' ' ' ' ' I ' 1 ~ - ~3 " ~ '.'; c ~ _, - X 3 ") -"-.;; ~ -- ~ ~ -~- 2 .... A.. ..,_ 78{3.3.9
6. Special Cases. When n =3.
tk.k'JE(nl (1) D
(6 1) (l,:,)E(3 ){ b b b . I • ) . (1) D a, l• 2• 3,C,C ,-,XpX2,X3
=Fe (a,a,a,b,,b2 ,b3 ; c,c' ,c' ;x1 ,x2 ,x3 ) T0 .e- T2 = 1, T1 + T3 = 1 .
( 6.2) (ldf E~)(a,bl ,b2 ,b3 ;cl ,C:z ,C;3 ;xl ,Xz ,X3)
= Fl3l(a,bpb2 ,b3 ;c1 ,c2 ,c3 ; x1 ,x2 , X 3 ) -'-- .,.. <1.
lt.k'lEl"l (2) [)
( 6.3) (IiliE};)(a,a' ,-,bl ,b2 ,b3; c; XpX2 ,X3)
=--P3 (a,a',a',b 1 ,b2 ,b3 ;c,c,c;x1 ,x2 ,x3 ), T1 +T2 =T1T2 , r2 =r3 •
( 6.4) (\~{Egl(a. 1 ,a2 ,a3 ,b1 ,b2 ,b3 ; c;x1 ,x2 ,x3 )
= F}/l(apa2 ,a3 ,b1 ,b2 ,b3 ;c;x1 ,x2 ,x3 ), lx1 j < 1,lx2 j < 1,jx3! < 1.
119
.6 - 1i.21E\3l( b ) l .t>J Hi c Pr>a2,a3, ;c1,C2,c3;xi>xz,X3
= F,l3l(b,a1 ,a2 ,a3 ;cpc2,C3 ;x, ,xz ,x3) lx1I + \x2I + lxsl < 1.
(6.6) ,a2 ,-,b;cl ,Cz.C3:X1 ,X2 .X3)
=FE b,b,a1 ,a 2 ,a 2 ;c1 ,c2 ,c3 :x1 ,x2 ,x3 } +(./;:;+JG) = 1 .
rn. b" ,-; C:, C:z ,C3; X: ,Xz ,X3)
=FE (a,a,a,b,b' ,b';c1 ,c2 ,c.3 ;x1 ,x2 ,x3 ), ( r- r·) + :Jr2 +JG = 1.
(6 . .8 IL2)Fi3J( b. "'~ AC a, "'v.:: ; C: ,Cz 'C3; X1 ,.'.Cz ,X3)
.c3;x:,X2,x3), !x1l+lx2l+Jxsl < 1.
'b.:3; c, 'Cz' C3; Xi' Xz' X3)
= F_!3 l(a,b1 ,b2 ,b3 ;c1 ,C2 ,c3 ;x1 ,X2 ,X3), + \x2I +jx3 1<1.
6.10) (a,b1 ,b2 ,b3 ;c,c' ,-;x1 ,x2 ,x3 )
= F0 (a,a,a,b1 ,b2 ,b3 ;c,c' ,c';x1 ,x2 ,x3 ), r1 + r2 = 1, r 1 +1;i = 1.
(6.11 ;c:x1 ,x2 ,x3 )
•3'{ = F3 1 a 1 ,a2 ,a3 ,b1 ,b2 ;c;x1 ,x20 x 3 ), < < l,jx3I < 1.
(6.12) (l.3
) F~~(a1 ,a2 ,-,b1 ,b2 ,b3 ;c;x1 ,Xz,Xs)
=F5 (a 1 ,a2 ,a2 ,b1 ,.b2 ,b3 ;c ,c ,c ;x1 ,x2 ,x3 ) r1 +r2 =r1r2 , r1 =r3 .
(6.13) b1 ,bz ,b3 ;c1 ,Cz,C3 ;x1 ,Xz ,x3)
= F.!31(a,b1 ,b2 ,b3 ;c1 ,c2 ,c3 ;x1 ,x2 ,x3 ), lx1l + \x2\ + lx3J < 1.
(6.14) (l,l) Fgl(a,-, b', b; CI ,Cz, C3; Xi, Xz' X3)
120
= FE(a,a,a,b,b' ,b';cl ,C2,C3;X1,X2,X3), rl + (.;-;:; + -J13r=1. Here FX31 ,F~.3l are Lauricella's series [ 14], while Fc,Fs,FE are
hypergeometric series of three variables defined by Saran [16] already
Conjectured by Lauricella [ 14] and r1 ,r2 ,1;3 are associated radii of
convergence of the triple hypergeometric series. 7. Convergence Conditions.
7.1 Convergence Conditions for ik(~·]E~).
F u1.k')E!n1 or ui D ,
(a 1 ,m1 + ... + m" Xb1 ,m1 } ... (bn ,mn J 1 1 Am, .... m,,
(c,m1 + ... +m.1:Xc',ni1:~z + ... +m1; ,m,,_1 -'- ... -"-m.n}m 1 ~···m"'.'
l:Sk ::;k':Sn.
Therefore
An11·····'11i i.Ill:""!"l.1n,.1··-·11i,, f( m )= A i _m.,, ... , "' m,,. .. ,m,,
r (a,m.J + ... + rn,_! + 1n, + 1 + m,_1 ..;.. ... + rnl: ..;.. ... .,.-
L (c,1ni + ... +m1_1 +1n, +l+mi-c + ... -m
(b 1 ,m 1 ) ... (bi-I.mi-I Xb, ,m, + lXb,_1 , ---- ~ - - ----· ------- -··-
( ) I ! ( . l\! . " ' ... m 1 ••••• ni1_1 • m, .,.. 1-m,_1 •.•• m"'. l
(a,m1 + ... +m,,Xb1 ,m1 ). .. (b",nz.J - m.: m.:
(c,~ 1 -+ •• ~~-;;;~Xc~,m1,~~-~ ... +mr. Xc'',mk-l -c- ... -,-mn)
(a+ m. 1 + ... + rn11
Xbm + nz,}
(c+m 1 + ... +m1 + ... +mhXm, -'--l)
Hence
( )- . a+ E (m 1 + ... + m 11 X_bm, + rn, E)
~i ml , ... ,mll - !~ ( ~ ( ' ' . ' ' ·x ~ ' . - C+c m 1 -r ...• m,-:- ... ,mk) nz;cTl)
(m1 + ... +m11
)m1
(m1 + ... +m1 + ... +m.1,)m1
1 m 1 + ... +m1;
m1+ ... +mn (which is independent of i) Therefore, r, q>;{m1 , .•• ,mJ
121
Hence r = r. = ... =re 1 1 .....
{1 s i s k)
Similarly,
r, - m~-1 .,.. ... 7- m,1;-
, m1 - ... -'-mn {indpendent of i)
= r,:-: = ... = rk . s i s k')
.Al Jn;.· _;_ .M.l!SO r, = "+1 · · ·· - m~ m- - ... -:-Tri.,
(Independent of i)
= r _, - ... - r,, ::; i :Sn)
Thus r1. + r 1, - r (m. + ... -'-m,)-(m,. __ 1 _;_ ... +tn;.-)+(m1.- .. 1 + ... +m)
i '• , . -~ ~ • ll
m1 + ... +mn
L iRequired condition)
where r. , ... J,, are associted radi of convergence.
required Codition for Convergence of is given
,.,. '/) -'iooiJ
1 l 1 -+-+-=1. rk rk. rn
Convergence conditon for 1 is given by
( 7.3) I . ·2 .,.: ~ ... - r" ) -'-
-- _;_ \12 . i ;-- .-•·; ' ... --:- _;_ ! •r. _;_ ' 'r r 1 "·\f' !:: -1 , ... T 'V n ' = .
Convergence conditon for ti:):' is given by
:? :2
( 7.4) 1 ; ~
,Jr1
1 1 1 1 1 -'--=1. ..,. -- ... ' i -V r.1; ,_/rt:-1
,--r1:·--..1 rn
Convergence conditon for 1
is given by
(7.5) rk + rk. + r 1,,.1 + ... + r11
= 1.
Convergence conditon for !kXIF1~~ is given by
( 7.6) 1 1 1 1 -+-+--+ ... --'-- = l, rk T1::· Tk-~1 rn
122
• tk k'\ In I · · Also Convergtnce cond1ton for · ·· FcD· is given by
(7. 7) ( ~ ~··?
rk + rk+1 + ... + rk. + -V T;,.,.1 + ... -;-.Jr,, r = 1.
We can also derive (i) Fractional Derivatives (iiJ Fractional Integration and (iii) Analytic continuation and multidimensional integral transforms etc. Also applications of these functions can be shown in (i
heat conduction problems, Electrostatic Potential problems and in other similar problems of physical sciences.
REFERENCES [1 J R.C.S. Chandel, On some multiple hypergeometric functions related to LauriceHa's
functions, .J/ianabha, 3 ( 1973), 119-136. [21 R.C.8. Chandd and A.K. Gupta, Multiple hypergeomerric functions related to LauriceHa's
func.;wns, Jnaniibha, 16 (1986), 195-209.
13] R.C.S. Chandel and Vishwakarma, PK., Karlsson's multiple hypergeomet:ric functions
and its confluent forms, Jiianabha, 19 1989), 173-185. 4] R.C.S. Chandel and PK. Vishwakarma, Fractional integration and integral representation
of Karisson's multiple hypergeometric function and its confluent forms, Jniiniibr..a, 20 ( 1990), 101-110.
[5] H. Exton, Certian hypergeometric functions of four variables, Buli. De Soc .• \fa.th. Grece, NS., 13 (1) (19721, 104-113.
!6l H. Exton, On two multiple hypergeometric functions, related to LaurieeHa's
Jnani'bha Sect. A., 2 (1972), 59-73. 7 H. Exton, Multiple Hypergeometric Functions and Halsted Press iEl!is
Harv.0od, Ch1..:hesterJ, John Wiley and Sens, :'.-;ew York .. Sydne}· and Tnrnmo. 1976.
[8! Karlsson, PW., On intermediate Lauriceila Functions. 211-222. 9 G. Lauricella, Sulle Funzioni ipergeometriche a
7 (1983), 111-158. va:ic.bi!i. Rer;d. Circ. Mat. Palermo,
10] S. Saran, Hypergeometric Functions of three variables, Gc.nitc, 5 \1954), 77-91. I 11 J H.M. Srivastava and M.C. Daoust, On Eulerian integrais associated with Kampe de Feriet's
function, Publ. Int. Math. (Beograd} N.S. (9)(23), (1969), 199-202).
[ 12J PK. Vishwakarma, Multiple Hypergeometric Functions and Applications in Statistics, Ph.D. Thesis Bundelkhand Univ. 1992.
123
Jnanabha, Vol. 37, 2007 (Dedicated to Honour Professor S.P. Singh on his 701
h Birthday)
CERTAJN MULTIPLE INTEGRAL TRANSFORMATIONS PERTANING TO THE MlJLTIVARIABLE A-FUNCTION
By
V.B.l.Chaurasia and Pinky Lata Department of Mathematics
University of Rajasthan, Jaipur-302004, Rajasthan, India E-mail : [email protected]
rReceived : August 10, 2007)
ABSTRACT The object of this paper is to establish two general multiple integral
transformations of the multivariable A-function (1981), as a Kernel product with and L~auerre polynomials respectively \.vith the gernal
pol:_v,,-:nomial!s and Several possible cases are also included. 2000 Mathematics Subject Classification : Primary 33C60; Secondary 33C70 Keywords: Multivariable A-function, Multivariable H-function, Hermite polynomials, Lagruerre pol:y11omails, Jacobi polynomials.
1. Introduction. Gautam and Goyal (1981) defined the multivs.riable A
function which is generalization of multivariable H-function of Srivastava and Panda f6l The difinition of multivariable A-function runs as follows:
A[z1 , ... ,Zr]=
1 \,r,{~ ·)_s, _srd d f'V,"1'···1Sr "°l ... "°r Si··· Sn ... (1.1) (2m:o)'
where ro = J=l ~L · - LL n r(d~i) - DV)si ITT
e. (s: \ = j=l . . j=l ! . '} c.
rtr(1-dy) +DY
+ C(i)S·) ) l,
) fr Vi= 1, ... ,r,
... (1.2) -cUls.·) J l
j=p,+l j=i-:-:--l
r . . j ~t ( r \ 1 - a . + ' A (i) s in r! bi - ' B(i) s. I
J L.. ! !; : J L.. J l;
""'( ) - j=l \ i=i . ) J=l \ i=l ) '*' S1 , ... ,Sr - ------------'----------
. n'.. f A{il nc r(1 b . *' B(i) i a)-£.... ~jsi I - j..,L,, jsil
j=;.+1 i=l j=p+l \ i=l )
!.
TI ... (1.3)
124
Here ~t,l.,v,C,p 1 ). 1 ,v, and ci are non-negative integers and all a i'bi, B(il , j '
are complex numbers. The multiple integral defining the A-functon of r-variables
converges absolu'tely if
;; =0 '
Ylz > 0,
and jarg(;;i )zk/< 11rr/2,
where
t' .
t:. - fl !.:ifi) "':l - fl..l. '.
. I J=l -
c Il J=l
fI J=l
1(D_,··ti
j J=l
; C C C. t'
-::' =img! 'A(ti_' B(ii+ 'D\i)_ 'c(i) i '"=t l L 1 L 1 L _; L J i'
LJ=l j=l j=l j=l J
... (1.4)
... (1.5)
... (l.6i
... (1.8)
TJi = ReJ1
~A (i) - f' A (i) + ~ B(_i) - f- B}i) .!.. f D'/1 - ~ D\i) +~ctn - ~ cUl L.1 L.1 L1 L. -'-<o L.- L1 L1
Lj=l j=l j=l j=l J=l J=l j=l j=i
Vi = 1, ... ,r.
If we take all A1U)s,B~ils,C~ 1s, and D\ll as real and u = 0, the A-function J ,! •
reduces '.:o multi·variable H-function of Srivastava and Panda f1976 f. -
Srivastava[ 4] introduced the general class of polynomials (see also Srivastava and Singh[7])
[fl. u] ( n) S il [ ] _ ~ - I-' ka A k n _ 0 1 ')
u z - .L.. k' f3,liz ,1-1 - , ,~, ••• , k=O ·
... (1.
where a is an arbitrary positive integer and coefficient ·'\-Lk (J3, k ~ 0) are arbitary
constant, real of complex. 2. The Main Results.
J:r f"'' .· -1 cr -1( ' \)CJ i}i (. Cl ·Cl r"
(i) ··· x' ···xr k1x>' 1 +···-'-kXP'. SPinkx~'-"-···+kx"".· c .Q 1 r 1 ' r r . a; ·1 .. 1 1 · · r r ,
Hm,01! r(k xp' +···+k x>'r ~(e,,E,), .. .,(el''EP P,q '" 1 1 r r A ( a ., ) (g ., ' ·1 b I '1 I , ... , </ ~: '!
Z1 .... Y1 :dx- ... dx
_z,Xr
[6 al , ... ,kr) )' --.......,
k=O ~ -n,t: ~4:.~:~::/;;~~~:;~-:~~;~~:~\"~.'.~:r .c,
c ... ~ 0 1 .... :Y,-1;_"
-!S-hk)-, :Z1i;; .... ,N.·;;j
;N,s, .... .J'(~. J, :·
{a :ir ...... 4.'.·-t :!: .C
l,b :B ..... B :id . . D'
where
X.= ~ • f""_,. '"". 'l -- ... -kTXT .
8=u+CT1 + ... +Gr
k) { )-1k-CT. C k-G n , ... , r, = CT1·--Vr. 'l · · ... r ·
N- =ni + i
and
Z - - ;:-:Y, I • .. i - ,c;i"::i R1
Z1
. Z,.
125
... (2.1)
. .. (2.2)
. .. (2.3)
... (2.4)
... (2.5)
. .. (2.6)
The above integral formula {2.1 l is valid under the following sufficient conditions:
(a) ki>O, p, > 0, ni 2': 0,
(b) Re (a.)> O,i = 1, ... ,r and
0 .__,. . fl l > , ·;l,J E ( ,. .. ,r,1,
r . .
R (s) ) ·r.,,r · min lR ( 'JI e . .· > - "---' ~ , io i - Ls.i:sm i e g_; !' i::=l
h 8 . Jo~!1d(i) iDU) ')l . 1 w ere i =mmp-=,) / ) ,),J = , .. .,µi,
(c) m, p, q, are integers such that 1::; m::; q and p 2': O,c: 1 >0
... (2.7)
... (2.8)
126
p q
(j=l, ... ,p), YJ >O(j=l, ... ,q) n 1 = L:sJ-LY1 <0, j:::::.1 j=1
Ill 4 p
Oz= LYJ - L YJ - L81 > O and larg(s~< 1/2 02;;:, j=l j=m+1 f=l 1
... (2.9)
(d) ~·"are arbitrary constants, real or complex and_ f),k 2 0.
(e) Conditions corresponding appropriately ( 1.4) through (1.6) are satisfied by
each of the multivariable A-function occurring in (2.1). Here H;'.~0 [z] denotes the
familiar H-function of Fox ([3],p. 408, see also [5], p.
(ii) f'1: f"" G -1 G 1( \J Jo ...• Q Xl i •. ·Xr r- k1xf' + ... + krX~',) +···+krx~· r
Z1X1 ..,,
rl. u),'.)-t 1 ; ... ;;t ,/.,.
t ,C:v1 ,c1 ; ••• ;ur ,Cr . .. dxr
~r
(- l)w -s [µ;u] (- R. ). • • = Y iir(k k ) L 1-';Ju:i A_. ,k.,.-::.~
( ) T' 1, ... , r £J.,~ ,n '. l kf •....
W . k=O '
A;1,i.+2:t1 1 ,i. 1 +1; ... ;p,,i., +1 v+2,C+l:v, +l,c1 .... ,v,+l,c
-S-hk;s1 P1.: ... :.:;. il
P.7" J:
[•, -S-h 1• .;: / • .;: " 11 ··A'·· ·A(rl) ·(1- / .;: .J.. ,.,+u,'::il P1, .. .,'::lrlPr \,aJ, J,. .. , j ·Iv' CT1 1 Pl>'::il
[1-S-hk + u + w;s1 I P1; ... ;sr; Pr],(bj;B'j , ... ,By) )l,C;
pi),
(.,. C ) • ·(l r;: /p .;: Ip },(-(r) c{rl)' 'j• jlv' ... ' -"'r r>~rl r> .. "J' j l·
' 1 .u,
(d' D' ·) · ... ·(d(r) D(r))' . J ' J l,cl' ' J ' J l,c,
. .. {2.10)
where L~l(z) be the Laguerre polynomials of order u and degree w in
z,wzO,~, >0,p; >0,s; >0,Re(cr;)>O,'v'; =l,. .. ,r
127
r i: 0 Re(S)> -I Si i ,Re(y) > 0,
t=l Pi ) ... (2.11)
\u(k1 , ... ,kr),S and i\ being given by 2.4), (2.3) and (2.8), respectively,
...,,, =zi i = l,..,r and conditions given by (1.4) through (1.6) are assumed
to hold for the multivariable A-function. 3. Proofs. To prove the main results, we take some assumptions for
r r
. ~ d "\"' _,,, d h . ' d ',,.Ul convenience L/i;si an L ... 5 1 's; enote t e r-terms sums L....,nisi an L....."'J si 1=1 1=1
respectively '7j = l, .. .,r.
Also, let
:1= C' .,. ..... ~&J -}. ~-1
. .. (3.1)
. . .... - . r z1 ~1 ~, l "· ) Aµ,1 •. µ1, .. .,>1r,1., l . Id. dx. ' . .:.. .:.. r. . . . . . . . ' .X1. .. r' .... k,....J::r vJC.v1 ,c1 , ... ,Vr~Cr , '
. , ! . . ~ I Z X 'or
r r J
... (3.2)
where the Xi are defined by (2. 2) and the fuction f is such that the multiple integral converges. On replacing the multivariable A-function occurring in (3.2) by contour integral given by (1.1), under the various conditions stated with (2.1), we find that
cl= , 1
. r- .. fe, (sl ) ... e_ (sr )<Pf s, , ... sJz1 51 ••• z/'
9 tr.. ~ · ' ' · \ •.. • ..... nm / i._ i_
fz ["' X er,-)'_, -l cr.-'~'s-1 .o ... ,0 . 1 . ._ ·. . .. x, - .. .. • "' X1 - + ... +k X P .• )L-n,s, r r 1
>' 1 ' .Lk x P·)d.,. dx lds ds 7 · • • ' r r '""'l · · · r J · 1 · · · r · ... (3.3)
Now we interrupt the innermost (x1, ... xr)-integral by using the following from of a known result [l,p.173].
~.cr.-l Xcr,-l(k p, ' k P- r f(k p k p )d dx ni(k k ) ·"1. . .. r 1 X1 . T . .• rxr ' \, 1 X1 l + ... + rxr r X1 ... r = T l •···· r
r(cr\ /p1} .. r(cr'\ /pr) r·" zcrllpl+ ... .,-cr,ipr+cr-lf(z}:J,z,
r( * I . . "'* P ) .o _Ci.1'P1-r ... Tv r r ... (3.4)
where lfl(k1 , .•. ,kr) is given by (2.4) and 1{;,~~{k;,p1 ,Re(oJ} > 0 then (3.3) reduces in
128
the following form
lf'(k1 ,. . .,k,.) j' ... ~ e ( ) e is \,r,(s ... s. )Y s, · · · Y. s, f'i. = . 1 Sl · · · r \ r !'*' , 1 ' 1 1 , 2mo ' • ( )
,. L .
( ~' I ) r( * Ip ) f f .,_ s-G-') n. s -l ( )d id d I'0_~1,P1··· ~ ,-11
r i.oz 'J......• fz ZJ 81··· Sn r(o- .. 1 IP1 + .... o-,.1p,.).
... (3.5)
where lfl(kp···,kr), Ni and Sare given by (2.4), (2.5) and (2.3) respectively, and
~' 1<1; Yi= z-k. L.·;
1"
1
l J ' '
r 0 '<Ii)
CT, = 0; + .t_,C,j Sj,
z=l 7j=1, .. .,r.
Now in the integral (3.5), we set
[ (e1,c1), ... ,(eP,cP)lsB
. cr m,O Z 'I , . f(z)=z Hp,q ~(g1,Yi), ... ,(gq,,q)J
... (3.6)
... (3.7)
and evaluate the z-integral by follmving familiar formula (when n=O), expressing the Mellin transform of Fox's H-function [5,p.311,eq.(3.3)]
n ( m ' a -flr(13J +B1sJUr 1- .r
•1lnm,n(zx):sl~ 1:1 ( B ·),TIP rfa; -'-A .L~ tD p,q TI r 1- 13 i - i8
• .!
z
J=m+l j=n~l
Interpret the resulting (s1, ... ,s,.)-integral as an A-function of r-variables, we will obtain the required result given in (2. Moreover to esttibhsh the other main integral (2.10), we can find relationship (3.5) in similar way and then we set
f (z) = zcr .·2xp(- yz)L~ l(yz )S~ [11zh] . . .. (3.
Ecaluate the innermost z-integral by using to a slightly modified version following well-known integral [2,p.292,eq.(1)]
Mf -yx L(al(vx},s}= r(a -s + m + l)f(s) v-s \€ m ' , l r( 1) ' m. a-s+ ... (3.11)
If we interpret the resulting multiple contour integral as an A-function of
r-variables, we will get desired results (2.10).
129
4. Special Cases. For the general class of polynomials, we take the case of Hermite
polynomials ([8,p.106,eq.(5.54)] and [7,p.158]) by setting sg [z] = zii:z Hµ 2 ~-Jl in
-y z
. which case a= 2, Aµ,k =(-it
Integral 1 (a): The result (2.1) reduces in following form
G"-1 0 Xl ... Xr. x l ~ - ... + kr."(r;: r
2 2
----;===l====, Hm,af,_fk x Pr -"- -'-k X p,) j (e1,s1), ... ,(ep,sp)l 1 { 1s i' p,q t'::~ 1 1 ' ·· · ' r r ( ) ( ) I
2.YTJ\k1x/' + ... +k.:c~- f" 1 g1,Y1 , ... , gq,Yq r . . ~
z1X1 [1312] (!3)!ka k,_-hk
... dxr =;-s"t'(k1,.··,kr)~ (13-2k)!k! 11 ':>
i zrX?'
.. · ~ ~ -, · · · · [1- p · I CT · : >" · f CT · .•. ,_ (r) /CT J Aµ,1 .. r. m.;., ,t." ... ,) . .t,,t., 1 1 1 S 1 J' '"' J J l,r i·~r~q,C~p-cl:ui"cl : ... ;c,.c.- [1-S +CT: N1 - n1 ,. . .,N,. - n,.],
[1- gJ -(S + hk}t J;N1·,1J ,. . .,N rY )1,9 ,
-ej -(S-,-hk)cj;N1£1 , ... ,Nr£
(aj;A'j ; ... ;Af) li,v;i;r j,CJ
·B · ·B(rl) ·(d D .) · ' :, ... , . ' ;'' ;' 1 J J ·l,C ,c1
1Z1 i \
l !Zr
valid under the same conditions as obtainable from (2.1). (ii) Integral l(b). The result (2.10) reduces in following form
J•:c 1·::r: .cr:-1 cr,-1\' '· _ \ +f}h.i2 0 ... _fi\ X1 •.. Xr h1X11-. + ... +krx/·,F
expr _).hx.P' , _,_ k X P. "bL(u)[)k X P1 . + k X P, h11 [l/2 · t r \.i'VJ 1.
1 • · •
1 r"' r 1 w ' \ i · 1 -r ·· · r r )J' 1
... (4.1)
130
.Hri 1 l
2 r;r-(·k x ,;l ~ .L k "· r lA:'.~'~i.:;~ ~;~~'.;;·· \/11\1?1 I ··· , .X I : , r - J
(-1 w '1-s) [13 fri)t ( l)k , , = I . lfl(k k \ """ \jJ • - P.'1-nk
(w)! 1 , .. ., rJ ;~0 (kft(j3-2k)1 T] ;
7 x ~·.., -1 l -
7 x "' -r T J
.A~i~i2 ~~~~\:; ~ ~ ~:.~·;~~-l~L:·:~ ~;r -S-hk;;1 P1; ... :,;r1pJ
dx1 ... dxr
[i -S-hP + u;:.;1 /p1; ... ;:.;r /Pr ],(aj;Aj; ... ; ·ll-G1 ~ \ ,....,1
- ... l - S - hk + U + w;; l P 1 ; · · ·;.; r Pr J'
(r'j ,C'Jl,u, ; ... ;(1-crr !Pr; :.;r Pr},
(dJ,DJ )1,c1 ; ... ;(d)r),nyl )l.c,
.c1rl
. )
valid under the same conditions as obtainable from
_,. v l
),
(2) ff we set a = 1 and Ai>.k = !3 / {V - ll;; , the general dass of polynomials
reduces in Laguerre polynomials ([8,p.106,eq.( 15, polynomials are given by
13 ( 13 + v) (- z )k L~l[z]= ,~lf.3-k (k)! .
and f7,p.159J) where Laguerre
(i) Integral 2(a) . The result (2.1) reduces in following form
rec rec CJ1-l CJ,-1{ p, p 'r' Jo ... Jo X1 . .. Xr \kl X1 , -l'. ••• + krxr ' j
(v)[ ( , P1 , .L • Pr r· l m,J >=( . P1 _,_ , Pr ·1' (e1't:i}, .. .,(ep,E £B T] k1X1 '···' kr)·r IHp,q l"' k1X1 ' ... -r-krxr ·.·( ' ) ( ' J 1g1,f1 , ... ,gq,
z1X1
... dx,.
_zrXr
- ;:.-5 -,":
il I. R , ( )" 11-'-r-V - ,. , ... ,krfli ._11_o-hk ~!, B-k. (- '' -= k=O'' · kf.
A;;~:~~~n~~~~-:~.:;~~~l;;~:~:·~ r ,Cr
'CTJ :;i ar···~~r Jui ,[l-gi-(S+hk}ri;N1yi,. . .,N,.Y)1,q
-S +a: N 1 -n1 , ... ,N,. -nr],[l-ei -(S + hk)c; i, ... ,N1r. i, ... ,N,.r. ) 1,p,
;A'J; ... ·
,B -
.C'J
,D·}
c(rJ) IZ1 l ' j . l,vr I : I D(r)) i . I'
, J l,c, IZ,. J '
valid under same conditions as required for (2.1). Integral 2(b)~ The result (2.10) reduces in following form
cr1-l cr,,-1 p 1 , , Pr J .. r x1 ... X,. (k1X1 --r- ••• -r-k,.x,. .
131
... (4.3)
_. ~'1 ...i.. ..l- D.- (u) V 7 ~'r .-L. -1- Pr {d- ;.t, ..:... ,..;.... ~ P, '; ( , ( ~ . exp[ l _k1 X 1 ..... krxr bL,, [, N 1 x 1 , ... , krxr Lµ t11(k1x1 ..••. k,x,. f]
dx1 ... dx,.
z,.~¥, ;,
• . ·r" !l fB+u 1\-11 .. -hk ( llL'v-s) k )'\'!' i~)i f .- . ' •lfl(k1,··., r ~l,,[3-k) (k,
w)! k-o.
.A:.1~~~~~;1:~~-~; ~~~-;~:~;t.:~ ~!, S hk . .c Ip · .i: Ip J - - v'"-::l ~ l , ... ,~r' r '
[l - S - hk .;.. u;.; 1 1 p1 ; ... ;.; r i p,.), {a J; A 'j ; ... ;A f) t ; (1- cr 1 i P1 ;E,l I Pi),
11-S-hk-'- -'-,.:I ... : ' Jib··B'·· ·B(r)) . l • ,u.w,-,1•P1,.·-,~rlPr, J' J, ... , j l,C>
132
(r'J ,C·j 1,v, ; ... ;(lc-0 r I Pr;~r I Pr}, (-r~:- 1 ,cfl )1
u
(d~ D' \.. . Jd(r) D(r) . r
j ' j ).1, Cl ' ... ' \' j ' j ... (4.4)
valid under the same conditions as obtainable from (2.10). (3) For the Jacobi polynomials ([8,p. 68, eq. (15,16)] and [7,p. 159]) by setting
st, [z] = p~s,t )[l - 2z] in which case a = 1 and
_ ( p + s) (s + t + P + l)k. Aii.k -\ p ) {s+l),,
(i) Integral 3(a). The result (2.1) reduces in following form
f"':· f"' cr,-1 v -1( P1 " r Jo ... Jo Xl . . .. Xr r kl XI + ... + krX/''
P.(s,t)[l 2 (k P1 , , k p, ·rlHm,of-(1
k P1 k Pr'} • f) - TJ. iX1 T .•. T TXT . ~ p.q l::;._ lxl _,,...···' rxr .
z.X.' i l
tc1 ,El ), ... ,(ep, E p (g 1 'y 1 ), ... '(g p 'y q
A il).:;t, ,i., , ... ,;tr ,i.r :d."C dx . .L~,C:v 1 ,c1 ; ... ;v,,cr ~ ·1 ••• .r
=!0-sl.f(k k )~lrp+s " 1, ... , r L
k=O P-k
zrXr
-"-t+k+s '...-hP. s k f' ~c
. ,., I ,. (r) : -pj/()j :r;j ()j,.-·,::;j ·0j ,[1-gJ-(S-'-hk}tJ;N1yJ, ... ,N,yJ
-S + 0: N 1 -n1 , ... ,1Vr -n, - ei -(S + hk}E J, ... ,N1 c: J, ... ,Nrr, )1,p•
(. ·A' · ·A(r)) ·(' C' .) · ·l'_(r! clri al· , 1· ' ... ' 1· , -r i , , 1 , ... , . ' i· ' ; · l,v - J ,v, \ _ .;
{b ·B' . ·B(r)) .rd' ·D' )' . . ·(·d(r) D(r) ~ )' j, .. ., j l,C'~ j• j l,c,'···•, j ' j
:z1 I , 1Zr I
valid under the conditions as required sufficienty for (2.1).
(ii) Integral 3(b). The relult (2.10) reduces in following form
r" r"' "1-1 a-i(k 11, k ~) 'r Jo ... Jo Xl ... Xr' 1 X1 + ... + rXr ' )
... (4.5)
.exp[ ' _j_ k P. hLi"'[·1(k . P: .+ ' k p, h j ... \ "rxr )J u: I ·• 1 X1 ' ... -r rxr )l
'·' 2 P;;'" 1-. T] !·"
r z x ~ 1 '/ _ I i i
~': k ,,.f A~,/.:u,.i.,,.,µ,.i.,I : d d Xl - ... - rXr ! ~ .. C "· ,._ ... , c ' . ' X1 ... Xr
, , - - --· ~ _,_ t ........ r , r- l I i z x ,;,J L r r
- (-l""f s) ( p ( f< . ' ( - qt · · p7S -'-t . • \ fwl! k,, ... h )'1 I[)· .,-k..,-s 1 h. \ r .. ·rf.::11<_k· r(k "=\)\!-' /1 k ' - \. )
.A:.~~~c~:t~-~-~ ;~.:1-~;:L_\.'~~ ~!~ S hk. ~ i . . ';- / p ]
- - ''-::1rP1~··•!'-Jrl r'
[1- S -hk ~ u:;1 ; ... :~, Pr },!a>A.'j ; ... ;.A.f) )1v;(l-01 / P1;~1 / P1),
c;; · u ·tr·': ··· · .: ,, l fb ·B', · ·B(r)) c·
i
-JU: - -.- -:-- .... :,"':'l; !~~~~;;r f ~TJ~' j' j , ... , j 11 '
-G:-'~i
1d' . D'. \, .C1 ; ... ; ~ J , J JL, ~
p,.i
,D\;)) 1 l,c,
,c(rJ J l
: J' valid under the conditions as required sufficiently for (2.10).
133
... (4.6)
(4) Ifwetake f3-+0 and µ=0 inresults(2.1),weobtainaknownresultobtained
by Srivastava and Panda [6,p. 354, eq. (LS)].
(5) On putting !3-+ 0 and µ = 0 in result (2.10), we arrive at a known result
obtained by Sriva:,"tava and Panda [6,p.354,eq.(1.14)].
ACKNOWLEDGEMENT The auto rs are grateful to professor H.M. Srivastava (University of Victoria,
Canada) for his kind help and suggestion in the preparation of this paper.
REFERENCES L.K Bhagchandani, Some triple Wruttaker transformation of certain by hypergeometric function, Rev. Mat. Hisp.-Amer. (4) 37 (1977), 129-146.
!2l A, Erdelyi., W Magnus, F Oberhettinger and FG. Tricomi, Tables of Integral Transforms, Vol. II McGraw Hill Book Co., New York, Toronto and London, 1954.
[3] C. Fox, The G and H-fonctionsas Symmetrical Fourier Kernels, Trans. Amer. Math. Soc. 98 (1961), 395-429.
[ 4j H.M. Srivastava, A Countour Integral Involving Fox's H-Function, lndianJ.Math., 14 (1972), 1-6. [5] H.M. Srivastava and R. Panda, Some operations techniques in the theory of special functions,
Nederl. Akad. Wetensch. Proc. Ser. A 76=lndag. Math.35 (1973), 308-319.
134
[6] H.M. Srivastava and R.Panda, Some multiple Integral transformations involving the H-function of several variables, Nederl. Akad. Wetensch. Proc. Ser. A, 82 (3) (1979), 353-362.
[7] H.M. Srivastava and N.P. Singh, The Integration of certain products of the multivariable Hfunction with a general class of polynomials, Rend. Circ. Afot. Palermo \2) 32 (1983), 157-187.
· [8] G.Szego, Orthogonal Polynomials, Amer. Math. Soc. Collog. Pub!. 23, Fourth Edition, Amer. Math.
Soc. Providence, Rhode Island !1975).
Jnanabha, Vol. 37, 2007 fDedicated to Honour Professor S.P Singh on his 701
h Birthday)
ON UNIFRORM T (C,1) SU.M.1\lABILITY OF LEGENDRE SERIES By
V.N. Tripathi and Sonia Pal Department of Mathematics
S.B. Postgraduate College, Baragaon, Varanasi, Uttar Pradesh, India
rReceived : September 18, 2007)
ABSTRACT The present paper deals with a theorem on uniform T( C, 1) summability of
Legendre series under very general condition. It generalizes a very recently known result due to Tripathi and Yadav (2007) on uniform (N,p,q)(C,1) summability of Legendre series under similar condition. It may be noted that the (N,p,q)
summability is a pa.i.--ticular case of the T-summability method. 2000 Mathematics Subject Classification : Primary 42B08; Secondary 42A24. Keywords : UPiform HC, 1 l summability, Legendre Series.
1. Introduction. An infinite series \Vi th Iu11 (x) with the sequence {Sn} of
its partial sums is said to be summable (C, 1) to a fixed and finite sum S(x) at a point x in an interval E, if the sequence-to-sequence transformation
c1 ,:z =~ f sm(x)
n m=l (1.1)
tends to S(x) as n~oo. [Titchmarsh (1959), p. 411].
Let T= J be an infinite triangular matrix v.ith entries an,k over reals or
complexes and
a,._k, = 0 for k>n.
The sequence-to-sequence transformation
.n
T,,(x)= .L>nJPi.(x) k=l
(1.2)
(1.3)
definies the nth T:mean of the sequence {c,~(x)} of the (C, 1) means of the sequence
{S,, (x }} of partial sums of the series IuJx), or the nth T(C, 1) mean of the series
:Lun(x) at the point x in the interval E.
If there exists a function S(x) such that
136
Tn(x)-S(x)=O(l) (1.4)
as n---;). OJ, unformly in E, the series I.un (x) is said to be summable T(C, 1) unformly
in E to the sum S(x). The Legendre series associated vvith a Lebesgue integrable function f(x) in
the range (-1,1) is given by
:.r.·
f(x)~ IanP11(x)
11=0
where
1 f 1 . . \,.J. an =in+ 2 __ /(x)Pn(xµ.-i::
and the nth Legendre polynomial Pn(x) is defined by the generating function
1 "' ~1-2xz+z2 = LPn(x)zn
n=O
We use the following notations
)= y 1,(t)= f{cos(8-t)}-f(cos8)
and
N (t)= ~a sin(k+l)i2sinkt·2 n ~ n~ .
k=l ksin 2 fi2
(1.6}
(1.
2. Main Result. In this section, our aim is to study uniform T{C, 1}
summability of Legendre series (1.5) under very general conditons by establishing
the following :
Theorem. Let T= (an,k) be an infinite regular triangular matrix with entries (an,k) as a sequence of non-negative reals, non-decreasing with respect to k and
nz
A = "\"' a,, k n.m L ··
k=l (2.1)
Let {p,) be a non-negative, monotonic non-increasing sequence of real
· coefficients such that its nth partial sum P,, ----,) CD as, n---;). :x:. Let i.(tl and
µ(t) be two positive functions oft such that lc(t), µ(t) and t1.(t)h1(t) increase
monotonically with t and
A.(n)P,, = O(~t(Pn)], as n ~OJ (2.2)
If
and
)\!f(u~ 2 u
= J 1.(l! t )p_ L µ(P7)'
u]du = o(l),
, as t----'? 0
as t-'?0, unformly in a set E defined in the interval (-1,1), where
11 =min[ arc cos u - arc cos(u-'- a)]
137
(2.3)
(2.4)
for u in (-1,1-a), a> 1; then the Legendre series (1.5) is summable T(C,1)
unformly in E to the sum f(
3. Lemmas. The following lemmas are needed in order to prove our main theorem:
Lemma!. : If is non-negative and non-decreasing with
b
. h f 1' i(n-hlt O(A ) ,k::::n ti en or 0::; a< b::; ='-, 0::;; t::;;;: and any n L., an,n-ke . = ' n,: ,
k=a
Lemma 2. If we write
N11
(t)= ~a sin(k + l)U2sinkt'2 L n.k _ __:.__~~=-=::..::~ k=l · ksin2 kt 2
then
for 0 ::;; t ::;; 1/ n
NJt)= nt2
1 n:;;t::::11.
Proof of Lemma 2. For O :::; t :::; 11 n,
n <In 1 __ , lsin(k + 1)t 2sinkr_12;i_ 1a ~, , ·" 1
- l n,k) , , ' k=l . kjsin 2 t 21
' .
( n \
=di LkJan,kl) \ k::.1
r n 1 I ,. I'
= oj (n)I!an,kl I L k=1 -
138
= O(n), as n ~ oo.
Using regularity conditions for T-method of summation for l/n s; ts; ri
IN (t~=l1 ~a sin(k+l)t 2sinkt 2[ n ~ ~ nA I
ksin 2 t 2 '
J~ {cost 2-cos(2k + l}t 2tj' l~an,k . . . . J 1k=1 ksm 2
t 2 i { 5
Ln costi2 II rLn cos(2k ~ l)t 2
s; a · +1 a · n,k k . 2 '2 I n,k k . 2 2 sm t 1 Jk=l sm t .
~ 0(1/ nt2 )+ ~ 1/ nt2 /R ~ a.,,e''"' 1' z.
=0 - 11+01'-( 1 \ r 1
') ·)
nt-) \, nt-
' n 2JllR°"' a .eikt ii~!!,;; 'i k=l
( 1 1 1 \I " '
=cO - +Ol-liR°"'a 'iin-H ·) ·) 1 ~ 11.11-.~:e
. nt- ', nt- // i:=l
= o(~ ·1 +a( AIZ,T J\ nt2
J nt2 '
as n~oo.
4. Proof of the theorem. Then n 1h partial sum 811
(x) of the Legendre series
0.5) at any point x in (-1,1) is given after a well knovn1 computation by
(4.1) Sn(x)-f(x)= 1 f.nf{cos(e-t)}-f(cose)~ .... 1 . . .
rc.Jsine .o sint;2 "m(n-l)tysm(8-t)dt+o(l)
rf"[ , ( )} ( )]sin(n+l)t , (. = o ft cos e - t - t cos e . . dt I+ o 1) o smr 2
=al rn l!f(t[ sin_(n + l)t l] + o(l) L Jo L smt/2 J
139
where,
11 = min[arc cos u - arc cos( u +
for u in a>O.
N O\V, the IC, l mean cr,, {x) of the Legendre series ( 1. 5) at x in (-1, 1) will be given by
1 rz-1
Gn\XJ-nx)=- L {S,Jx)- f(x)l nm=G
1
n
1 =
n
(tl .·n-1 ,
~<) sin(m+l)t rdt+O(l) • .L. 9 ~ ' ;
Slfll ~ --'' J
u(t)sin(n-l)t 2sinnt 2dt+O(l). sin2 t 2
(4.2)
we have T(C.1) mean T (x) of the sequence S (x) of partial · n n
sums given
=) ~
)-,;;:::::::-.,.
1.u(t'.(.~ a sin(k+l)t 2sinkt/2 l O() . A L n .k . 2 ~t + 1
,k=l ksm t 2 ;
= ~dt -'-0(1) ,, (say) (4.3)
where
Nn n . l =Ia . sm1k..,...1)t 2sin
" 7Lf::. _ :; P.=l ksm- t 2
2
Now, if we show that
Tn(x)- = (4.4)
as n .-+ w unformly in a set E then the Legendre series (1.5) will be summabile
T(C, 1) unformly in E to the sum f(x).
Let us v. ... ri te
I= Tn(x)- = )N n (t )dt + 0(1)
= fl. n + rn -"-0(1) .o .1 n
140
= 11 +12 + 0(1), say.
Firstly, we consider I 1. Now,
II1 I~ f ;! 'l l\j!(t AIN n (t ~dt
( ) fl/n ( v
= 0 n Jo JlJf t 1idt, using lemma 2
)ol A(n )Pn l . = O(n l k(Pn) J, usmg (2.3)
.t.(()P) l,sincenpn s,.Pn by the conditon on {pn}. k pl! j
= o(l), using (2.2)
as n -* Cl) , unformly in E.
Next, we consider 12 . Here,
!I2 1~ fq l\J1(tvl1Nn(tvdt
1 ' l; nl Ii Ii
= o(!) f 11 J\Jl(t ~ dt + o( ! n l/n t2 \\n
j\jJ(t). An_,dt in t2
= O(l/n).12.1 +0(1/n),!2_2 , say.
Now,
[ 1 { A(l/t )p,
12.1 = [i· 0 k(P,) J·ri 1' / •• {t.t)p_ ·. 1 d" + 0) . ' >-;;- [ i; n i k(P,) _, rs
=0(n2plrlc(n::n l+of .t-(n)pn rnr3dt k(rn) J L k(f'.'1) -
= Olr nA(n )Pn J- + o[ lc(n )Pn k(Pn) k(Pn)
=-O(n)+d n A n)Pn r 2 ( ]
L k(Pn)
= O(n)+O(n)
=O(n)
t-2
-2
(4.5)
(4.6)
(4.7)
so that
0(1/n as n --+ x , unformaly in E.
Lastly considering I 2_2, we have
12_2 =
=
l ~A- _dt
r: t.:- ·~·.
asn-+x,
unformly in E, on using (3.4), so that
O(l!n).!2_2 = 0(1/n
= o(l),
as n --+ x , unformly in E.
Combining (4.7), 4.81 and {4.9), we get
as n -+-::.u:., in E.
141
(4.8)
(4.9)
(4.10)
Lastly, combining 4.6! and 4.10), we obtain the required result in (4.4).
This the proof of our theorem. ACKNOWLEDGEMENT
The authors are very thankful to Professor L.M. Tripathi, Retired Prof. and Head, Department of Mathematics, B.H.U., Varanasi for his valuable suggestions during the preparation of the present paper.
REFERENCES
[ l] S. Lal, On the degree of aproximation of conjugate of a function belonging to weighted W(ll, ~ (t)) dass by matrix srnnmability means of conjugate series of a Fourier seires, Tamkang Journal of Matherna.tics:, 31 t4\ I 279-288.
(2] E.G. Titchmansh, Theo;i:.-o{Fun.c:tion.$, Hnd Edition, Oxford University Press (1959).
[3] V N. Tripathi and RS. Yadav. On Cniform \N,p,q) (C,l) summability of Legendre seriesVijnana Pariashad Anusandha..11 Patrum. 50 !3) July {2007). 243-252.
Jnanabha, Vol. 37, 2007 Dedicated to Honour Professor S.P. Singh on his 701
h Birthday)
FINITE M!G/1 Qu'ElJEING SYSTEM WITH BERNOULLI FEEDBACK UNDER OPTIMAL N-POLICY
Bv
M.R. Dhakad Government Women's Poly-technic College Gwalior-474001, Jrfadhya Pradesh, India
and
Madhu Jain and G.C. Sharma Department of Mathematics
Institute of Basic Science, Dr. B.R. A.mbedkar University, Agra-282002 , India E-mail : madhufma~ iitr.ernet.in, [email protected]
fReceir.,·ed : September 25, 2007)
.. IBSTRACT i;;estigates }vfG. 1 queue with Bernoulli feedback under an
According to N-policy, the server renders service only \Vhen
N customers are accumulated in the system; once he initiates service, continues to provide it till system becomes empty. The system size under steady state
conditions are determined by using generating function and supplementary variable technique. The analytical results for average queue length and mean response
time are obtained. The optimal N-value is obtained by minimizing the total operation cost of the system. 2000 Mathematics Subject Classification: Primary 68M20; Secondary 60K25. Keywords : 1\f/G/l queueing system, Bernoulli feedback. Supplementary variable. JV-policy, Cost analysis, Tota1 response time.
1. Introduction. Feedback queueing networks are important in many applications where the customer getting incomplete service may try to seek service
repeatedly till his service gets completed. Such situations arise in manufacturing,
data communications, telecommunications, packet transmissions, etc. Several
authors have investigated the feedback queueing systems in different frameworks; some of the recent works are as given below. Disney et al. (1984) derived stationary queue length and waiting time distribution in single server feedback queues. Vanden et al. 1991) studied the M/G/1 queue with processor sharing and its relation to a feedback queue. Sojourn time in vacation queues and polling systems with Bernoulli feedback was discussed by Takine et al. (1991). Rege (1993) discussed the M/G/l queue with Bernoulli feedback. Adve and Nelson (1994) gave the relationship
144
between Bernoulli and fixed feedback policies for the AfG 1 queue. Thangaraj and Santhakumaran (1994) analysed Sojourn times in queues \vith a pair of instantaneous independent Bernoulli feedback. Bhattacharya et aL 1995) studied on adoptive optimization in multiclass lv!iGJ/l queues ·with Bernoulli feedback. Takagi ( 1996J had studied the response time in 1WiG, 1 queues with service in ra.'ltdom order and Bernoulli feedback. Fujian and Yang 1.1998.1 discussed queue length performance of some non-exhaustive polling models with Bernoulli feedback. Choi. et al. (2003) discussed an M-'Gil queue with multiple types of feedback with gated vacatio:'.ls and FCFS policy.
Many res2archers have incorporated the concept of N- policy in queueing
system in which the server turns on whenever Nor more customers are present the system. Tang (1994) obtained takacs type equation in the ;.lfG'l queue
policy adn setup times. Piscataway and Choudhury ( 1997) considered a Poisson queue under Ncpolicy \vith a general setup time. Lee and Park l.1997) discussed a11
optimal strategy based on N-policy for production system with early setup. Jain (1997) has introduced an optimal N-policy for single server Markovian queue with breakdown, repair and state dependent arrival rate. _.\rtalejo 1.19981 gave some
results for the M/G/l queue under N-policy. Gupta ( 1999) considered N queueing system with finite source and vmrm spares. Hur and Paik, 1999) analysed effect of different arrival rates on the N-policy for lv!/Gil ·with server et al. (20J2) considered the N-policy MiG/1 feedback queue
rates. Jain and Singh (2003) have discussed an optimal
dependent M!Ek/l queue with server breakdovms. Jain
the state-
for redundant repairable system \vi th additional repairmen. Pearn adn Chai'lg
presented optimal management of the N-policy JIB~ 1 queueing system with a removable service station. Berman and Larson (2004) have analysed a quet: control model for retail senices having back room operations and cross-trained workers.
Several authors used the supplementary variable technique in Hokstad (1975) has applied supplimentary variable technique in the
queue. 'l queue.
Wang an.' Ke (2000) employed a recursive method using the supplementary variable technique to established optimal control policy at a minimum cost invohted.
The purpose of this paper is to obtain the mean response time and optimal
operating N-policy for finite MIG/I queueing model \vi.th Bernoulli fee.dback. The
rest of the paper is organized as follows. We provide notations and terminology related to model in section 2. In section 3, the probability generating function
technique to obtain steady state probability distribution of the number of units in
the system is discussed. The supplementary variable technique by treating the remaining service time as supplementary variable is used. The explicit results for
145
the average number of customers in the system and the mean response time are determined in section 4. The optimal operating N-policy is also stated to minimize the total el\.o-pected cost per customer time. In section 5, we deduced some particular cases match \v"ith earlier existing results. Finally conclusion is drawn in
seciton 6. 2. The Model Analysis and Description. Consider M/G/1 Bernoulli
feedback state dependent single serrnr queue v.ith general service time distribution and finite capacity K under .V-policy. The custom~rs arri.,,-e at the system according w Poisson process with arrival rate i. and th~ system permits no arrivals when the number of customers in the system is reached to K. The service time of each customer is independently identically distributed (i.i.d.) random variable with
distribution function B(x Xx 2: 0). A single server renders the service according to
FCFS discipline. As soon as the system becomes empty, the server turns off and may not operate S customers are accumulated in the system. A customer departs from the system after completion of the service with probability cr. If the service is not successfully completed then the customer again joins the system and is cvcled into senice \Yith probability (1-cr). We apply supplementary
variable 'L>t:X111uy;
time. introducing the new variable X denoting the remaining service
The following notations and terminology are used to formulate the mathematical model :
CT
N
x w pd{
LST
-traffic intensity such that O < 0 ::::: 1.
- Bernoulli feedback parameter. - Threshold parameter in turn on policy. - remairing senice time for the customers being served. - expected 'lvaiting time of the costomers.
- probability density fonction. - Laplace Stieltjes Transform.
- probability generating function.
B - service time random variable. B(x) - service time distribution.
- pd{ of service time distribution.
- LST service time distribution.
e )- order derivative of service time distribution with respect to e. H H(x) h(x)
Free)
- senice time random variable in Bernouli f cedback queue. - service time distribution of H.
- pd{ of service time distribution of H.
- LST service time distribution of H.
146
H*(i>( e )- ith order derivative of H with respect to e. p 0_0 (x) = Pr (the system being empty at time t)
Po,n (x) = Pr (there are n customers in the system at time t when the server is turned off, where n= 1,2, ... ,N-
Pi,n(x) =Pr( there are n customers in the system at time t when the server is turned
on and working, n = 1,2, ... }.
P\,n(8) = LST of Pi,n(x).
N-1
Go(z)= LZnPo,n,pgfof Po,n(x). n=O
K
G1 (z,O)= Iz"p;..,(O),pgfof p1,,,(x). I;l
K
o;(z,9)= °Lz"p;,,,(e),pgf of p;_n n=l
G(z) = pgf of the number of customers in the system.
Let us assume that the customer is feedback the end of the queue and return again and again till the service is completed. The total service time
the customer is the time from the initialization of service upto deparnrre
from the system. It is noted that (cf. Kumar et
crB'(e) H"(e)= {l-(1-cr)B' 1
Thus? the mean service time H'\1 1(0) and t:he second moment H 12 '(0) of the
service time of customer is given by (cf. Takine et aL 1991)
H'llJ(O)= -B''(ll(o) CT
H·r2i(O)= B'\21(0) .c. 2(1-cr) 0 . ()2
... (2J
3. The Analysis. We now proceed to analyze 1'.rfG; 1 queueing system \vi th
Bernoulli feedback under N-policy by using gererating function and supplementary
variable techniques, treating the remaining service time as supplementary variable
X to obta :n steady state results of the model under investigation. Let us define
Pi.n (x,t )dx =Pr .{Z(t) = n,x < X(t)::; x + dx} x 2'. 0, n=l,2 ...
Pl,n(t)= f: P1,n(x,t)dx,n = 1,2, ...
The equations that are governing the model are as follows:
d dt
d dt
c
=
,~
L
--- v .. ct ex"'·
c c ,::;t ex
-'-Pu (O,t)
(th· t ). lsnsN-1
l = -ip 1 ;(.;;,t).,... p 12 t
. . (' ~ . t ,l = -i.P:.," X, l ip\x, Pi.n-1 (O,t)h(x), 2snsN-1
c c ;"' -::::- •p, \· 1
X i""l- · d ere ... \ •. - t I- ipLV-l (x,t )+ ip0_N-l (x,t)Pi.N.;.l (O,t )h(x)
147
... (3.1)
. .. (3.2)
... (3.3)
... (3.4)
... (3.5)
;:_
= t ,_ t ).:.. Pi.n-i (O,t )h(x),N + 1:::; n:::; K -1 ... (3.6) 1Ct C:T
a ex .t)
steady state, we use the follovving notations
Po .• , = nm Po.n (t ), r........+Y:
n=0,1,2, ... ,N-l
Pi.rz = lim PLrz (t ), t_,.z n= 1,2,. .. ,K
P1,rz =hm n=l,2, ... ,K. t~:c
Further let us define
D, ,. , (x) = J. ;~··-· -- -1 ) .
In steady state, the equations (3.1)-(3.7) reduce to the following:
0=
O = -t-o
d --d p .•
• "( l •
+ 1.,,t5,.pu, l:::;n:::;N-1
)- Pu (O,t )h(x)
d ' ' ) ' " . ( ) --p1 n(xl= /._p,..., (x -c-),p, n-1· (x}_;_pl "'-1'0)h x , 2 < n < N-1 d .ii... ~ , , ..L.H \ ..L. £ , .!• \ - -
x
... (3.7)
... (4.0)
... (4.1)
... (4.2)
... (4.3)
... (4.4)
148
- :x Pi_N (x) = -IP1N (x) + tPuv-1 (x) + lpo _v-1 {x) +Pu·_, ( 0 )h(x ),
- :x P1)x) = -i.pLn (x) + )pl.n-1 u i-:- ( 0 )h ( x), N + l :::; n :::; K - l
- dd Pi.K (x) = lcJJ111-1 (x) . x
Using equations ( 4.1)-( 4.2), we obtain
P11(0)= i.oPoo = i.oPo,,, l:Sn:SN-1
When we take LST of equations 7 \Ve get the following:
(le - 8 )p~ 1 (8) = p 12 (o )H' ( 8 )- Pu (o)
(!, -8 )p; a (8) = lp~_n-I (e )+ Pi.n+l (o )H" (El)- Pin }, 2 :s; n :::; N - l
(I, - 8 )p;1 .- (8) = lcp~_l'l-i (8)+ Pi.s-i (o )H' (e).,.. i.Po.s-i (O )H~ (8)- Pi.x (o ),
(/,-8)p~ 11 (8)= lp 1,,, (8)+ Pin-1(0)H \ti}- p 2 .,
-ep; K (8) = lcp~ K-i (8 )- Pix (0) Using (5.0),we get
AoPi.N-IH" (8) =Pu (o )H' (8)
Now using this result equation (5.3) reduces to
(!., -e )p;_N (8) = lp~·s-Je )+ PiN-1 {O)H (B )- P:.
Now, adding equations (5.1)-(5.5), we get
f . · )- /c(l-13) · )+ 1-H''(e) f ;SPL11 (8 -
8 P1K-1 (e .
8 ~P1:
Applying L'Hospital' s rule in (8), we get
K b K
I P1" (8) = _1_ LPLn (o) ll=l () ll=l
where b1
= B"(l 1(0)
1V-'-l.Sn.SK-l
8 - D-
... (4.5)
... (4.6)
... (5.0)
1-... tu.
... (5.2)
... (5.5)
j
3.1 Probability Generating Function Technique. The equations (5.1)
and (5.2l are diffcult to be solved recursively. In vie\v of this we proceed to obtain
;:;nalytic solution for Pi,n ,p;,n (0) (n = 1,2 ... ) in closed-form by using probability
generating function technique.
149
Using = 0,1,2, ... , .. :Y - we have
1 --"" G, (z l = -=-=-- P , 1 ", . 1-z ... , 10)
Multiplying 1 and 5.5 i by appropriate power of z and summing, we get the
foHmving equation
.. ,, ·-v-'\·--,) H(e) l·G 1 -0·l --sH,. H-/_L,fCT: ~~tJ :=: --- ;- ~\..C~ .-/_::, - lcH" (e )p0_0 + 1..zK p~'.K (e) ... ( 11)
~ov,· putting - ; . .z m 1.1 , \Ye have
G. ;·.z': Pu;· - - i2)
if • - • . : \ 1 11H (1.-1.z)iZ)-lf
. .. (12)
Substitutmg from 12) into (11), we find
I=----·- ___ 1 l -- . -i2L_ -H(e)IP0.o+l.ZKP1K
- i..z)-z 1 J -i..zl ... l
By Putting e = O in U3l, \Ve obtain c;(z,O) as
:.i:
G }=·l1~zsi
150
4. The Expressions for performance Measures. \Ve no>v, propose to find the following performance characteristics in exlicit form by using generating
functions tecnique as follows: (i) Expected queue length. The expected number of couscomers in the queue
is obtained by using (17) as
L = G'(zJiz~i
l~ N(N -l}v(a-pj-2Np(Ci-p)-N/.2a 2 H-(21 (0)
2kr- p)2
., . . - T - . ~ -r' + !X(K -1) P~,K (0)-2-/' Kp~_k (0 )-c- i_'° Kp~pf
where p:1l(o) and P;'.;f (O) are determined by using 17!
and H"(21=JB'i2J(o) 2(1-CJ)i -
L, + ·B-n CT . 02 I
where B'(2 l(o) stands for the second moment of the service time.
(ii) Mean response time. The mean response time is given by
E(R)= W +B"(11(0)= ~ +B"( 11 (0) A
where the value of L is given by (18)
(iii) Optimal N-Policy .
... (18)
., .. \
Let E[I],E[B] and E[C] denote the e2-..1Jected length of the idle period. the
busy peri0d and busy cycle, repectively, such that E[C] = E[J1 E[B]. As the length
of the idle period is the sum of l\T exponetial random variables each having a mean
of 1 //" so that we get
E[l]= N f~ .
The long-run fractions of time of idle and busy server are given as
E[I]_ ()- -E[C]-G0 1 -NPo.o
E(B]=C--(1,0)=·1' Np ')Po,o+l-.Kp~_K(O)-lp;1~(0) E[C J . - - \CT - p)
... (20)
... (21)
The number of busy cycles per unit time is obtained as
1
[- = ip EC] . O.D
we now consider a cost function in order to get optimal value of threshold
parameter N as given below:
E =ChL-rC, Ell\_._ ·EC'
~---{C +C )-1-E{C] . ' . d E[C]
where, Ch= Holding cost per unit time per customer present in the system.
Cr= Cost incurred per unit time for keeping the server off. ' C0
= Cost incurred per unit time for keeping the server on. C
8= Start-up cost per unit time for turning the server on.
CJ= Shutdown cost per unit time for turning the server off.
1
151
... (22)
... (23)
Since E[C] is independent of the decision variable N, neglecting fourth term of
we get a new eost function per unit time to be minimized as follows:
! VI = C. . • ~·ll~ -_U;_,..CrNPo.o +CoNl aL-pjPo.o ... (24}
. _ dE{C(N): fmd the optimal value of threshold parameter N lsay fV), we put .-u · = 0,
yielding to
:.V~ =_.!_· C1-; -.JC.--'- C.oP ·. C_;; 2 \u-pl ...{25)
\Ve observe that is not an integer, the best positive integeral value of N . ' ... • --11 d' h al f "':""' is obtaineu roun ing t ie v ue o 1>1 •
5. Particular Cases. In this section , some special cases by taking appropriate parameter, will be obtained as follows
Casel: The M!G!l Model with N-Policy and Bernoulli Feedback. Putting
=it and k---)- :r.: i.e. the case is of infinite capacity system, we get results as
follows:
The average numbers of customers in the system and mean response times are
01 -1 . L= - 2 -;- CT
. ·)
Cv E'1
21(o) .c. 2(1~cr){B'(1l(o)}2 l cr er- J ... (26)
l '12
and E(R)= N ~1 7 261 + IB t:!!(O) + i.(l-0) /B ,11 (o):2
21. CT 2(0-p) G(G-p)· ... (27)
By changing the specific distributiobn for service time, the expressions for
the average· number of customers in the system and mean response time reduces
to
N-1 P p" exponential ---"""T"" ~ - -----
2 CT G(c;-p)
1 N -1 p p 2 (2-u) L = ' -2- -;- CT + -2-0-,--( CT---p-)' deterministic
~7\T -1 . p . p2 (3-u) ~'CT-;- 30(0-p)'
3 -stage Erlang
N -1 261 p
~+--;--+ p(a-11) exponential
E(R)= N-1. 261 . p ~;(1-rr)
21. -;---;--.,.. 2,ll(u-p)- 0,u(0-p)' . N -1 26, 2p p(l - 0) l ~ +--;;- + 3p(u - p) + Ci~l(G -p),
detenninisi tc
3 - stage Erlang
... (28)
.. .129.'
Case 2 : The M!G!l Model with state Dependent Arrival Rate.
H (8 h B (tl) and N--+ 1, then our model converts ro Gl
depender:t arriv.:.;.l rate which was studied by Gupta and Srinivasa
Their results are derived directly by putting above ;;a.:riable.
state
1996a).
Conclusion. In this study, we have considered an optimal N-policy for 1\'J/
Gil finite queue with Bernoulli feedback. The analytical expressions for the average
number of units in the queueing system are established by using generating function
and supplementary variable technique in steady state probabilities. The cost
analysis is helpful for the system designer in determining [he optimal 'rnlue
threshold parameter so as to minmize total expected cost. The analysis of queueing
system operating under N-policy and Bernoulli feedback mechanism, p:ro'l.r:ides
unified treatment in many real life congestion problems encountered in computer,
commun;cation -:;ystems, manufacturing, production and distribution process.
REFERENCES i 11 VS. Adve and R. Nelson, The relationship between Bernoulli and fixed feedback
Gl queue. Opei: Res., 42 No.2, ( 19941, 380-385.
i 2 ! J.R. Artalejo, Some results on the MIG/ 1 queue with N-policy. Asia J.
for the Mi
Res., 15No2
153
!l998i. 147-157. [ 3~ P. P. Bhattacharya. L. Georgiadis and P Tsoucas. Ploblems of adaptive optimization in multiclass Mi
Gil queues -,vith Bernoulli feedback. Maths_ Opa Res., 20(2) (1995), 355-380. l 4 - 0. Berman and RC. Larson, A que :1ei ng contra[ model for retail service; having back room operations
and cross-trained workers_ J_ Com. Opu: Res .. 31 <2004), 201-222.
i5l G. Choudhu0: A Poisson queue u:-ider with a general setup time, Ind. J Pure A.ppl. Math_
28 1997- - 1595-1608. 61 3.H. Choi. B. Kim and S.H. Choi .. -\n 51 G l queue with multiple types offeedback gated vacations
, J. Com. Oper. Res., 30 ; 20G;L l..289-1309.
D. Koning and\' Schmidt. Stationary queue length and waiting time distribution in
server feedback queues. A.de-. Appl. Prob .. 16 1984 1 437-446. _ 8 L Fujjan and 0. Yang. Queue length performance of s0rne non-exhaustive polling models with
BernouHi :feedback. IEEE J1ilitan· Communicaiions Conference. Proceedings. New York, 3 (1998)
888-892. f9] s_:\t Gupta. N-policy queueing system with finite source and warm spares. OPSEARCH, 36 No.3
1999'. 189-217. :wJ P Hokstad. _.'..._ supp~eme:--.-cary variable technique applied to the M!G!l queue, Scandinavian J.
St-at .. 2 19-;5, 95--95" 5. Hur ''-'·d. S.ci. Pai.k.. Efiec: of different arriYal rates on the N-policy of MiG 1 with server setup
23 No.4. 1999 . 289-299.
12 sen·er :'.1.farkoYian queue with breakdown. repair and state
13 No. 3 119971. 245-260.
redundant repairable system with additional repairmen_ OPSEARCH. 40
No.2 '14, M. Jain and P Singh, Optimal N-policy for the state-dependent ME,, 1 queue '<vith server breakduwns_
AmeriaznJ. J!fath. J1anag. Sci./USA; 12003J,1In press)
15: B.K. Kumar.D. AriYudainambi and A. Vijayakumar. On the :\'-policy of M.G l feedback queue with varying arrivpj rates. OPSE.4RCH, 39 No. 5 &6. ;2003 296-31-L
16] H.W. Lee and J.O. Park, Optimal strategy in ,\·-policy production system with early selUp. J_ Oper.
Res.Soc..48 >io.3 1 1997:. 306-613. l 7i W.L. Pearn a.:.d Y.C. Char::g. Optimal management of the N-policy J1iEi:, 1 queueing system with a
removable sere.ice station: a ser::siti>ity investigation, Com. Ope1: Res_ 3112004), 1001-1015.
18l K. Rage. On the !.HG l q:...ieue ·,•,ith Bernoulli feedback. Oper. Res. Lett., 14 No.3 ( 1993), 163-170.
19. H. Ta.k.agi. A note on rhe respJr:se rime in Jf G l queues w·ith service in random order &i1d Bernoulli
feedback_ J. Opa Res_ Soc 39 No. 4i 1996l. 486-500.
~20 D.C. Tang, A takes ty-pe equation in the JIG l queue 'Nith N-policy and setup times. Int. J. Inform.
Jfonag. Sci.5 No.3 ! 1994 l, 49-54.
[21] T. Ta.kine, H.Takagi and T Haseawa, Sojourn times in vacation queues and polling systems with
Bernoulli feedback, J. A.ppi. Prob .. 28 ! 1991 '· 422-432.
,22] Y Thangaraj and A. Santhakumaran, Sojourn times in queues with a pair of instantaneous
independent Bernoulli feedback. Optimization, 29, No. 309941, 281-294.
·2;3_ Berg TL. \'anden and O.J. Boxma. The Jf G l queue with processor sharing and its relation to a
feedback queue. Queueing Syst., 9 ! 1991.1 365-401.
[24] K.H. Wang and J.C Ke. A recursive method to the optimal control M/G/l queueing system with finite capacity a.rid infinite capacity. App. ;\1ath. Model., 24 (2000), 89-914.
155
Jnanabha, vol. 37, 2007 rDedicated to Honour Professor S.P Singh on his 701
h Birthday)
g-REGlJLARLY ORDERED SPACES Bv
M.C. Sharma and N. Kumar Department Mathematics
N.RE.C. College, Khuja-2031.31, Uttar Pradesh, India e-mail : [email protected]
fRecei1.:ed: Sovember 15, 2006; Re·vised : October 10, 2007)
ABSTRACT In the present paper, \Ve introduce the order analogue of g-regular space
and prove some results about g-regular space to g-regularly ordered spaces. 2000 Mathematics Subject Classification: Primary 54F05; Secondary 54E35. Keywords : Decresaing g-dosed. increasing g-open, T112-ordered, g-regularly
1. Introduction. A topological ordered space is a topological space equipped
of topological ordered space was initiated by Nachbin
ue11lie::, a topological ordered spaces to be a topological space if it is also equipped with an order relation. The order may be a preorder (i.e. a reflexive and transitive relation) or a partial order (i.e. an antisymmetric preorder). Every topological spaces may be regarded as a topological ordered space equipped -..vith a
discrete (trivial order·~·, where x ~y iff x=y. The concept in topological space was
introduced Munshi [2]. In this paper we introduce the order analogue of g-
regular space 'Ne prove some results about g-regular space to g-regularly ordered spaces.
2. Preliminary. Let Xbe a set equipped \vith a partial order'~'. For yEX,
the set {xEX: x~y} ·will be denoted by[~, y} and the set {xEX:y ~x} will be denoted
by . For any subset A of X.
and if
i(A) = d(A)
u
.__,
~J:aEA}, and
fI~.a]:aEA.J.
Obviously, A-;::;;; i(A) and A -;::;;; d(A), If A=i(A), then A is said to be increasing
then A is said to be deccreasing. In other words is A increasing iff
x ~ )' and x E A => y E A, and A is decreasing iff x ~ y and y E A ::::> x E A. The
complement of a decreasing (increasing) set is increasing (decreasing).
156
Let (X, ~,Tl be a topological ordered space. For any subset A of X, let
D(A)
DA>
n {F : F is a decreasing closed set containing A},
n { H : H is a increasing closed set containing A,,
Df'(A! = '--' { G :G is a decreasing open set contained in A,,
[D(A.) = u {M :JJ is an increasing open set contained in A}.
It can be easily seen that DIA) (l(A)J is the smallest decreasing (increasing;
closed set. containing A and D 0tA \[O\A)) is the largest decreasing (increasing}
open set contained in A. 2.1 Lemma [4]. If A be any subset of a topological ordered space X and if D(A.),
[(A), Do(A), [D(A) be as above, then the following hold :
(i) X-D(A) = J.0 (X-A
(ii) X-J(A) = D 0 (X-AJ,
\iii) X-lo(A) = D (X-A),
(IY) X-D 0 (A) = l (X-A).
3. g-Regularly Ordered Spaces.
3.1 Definition. A subset A of a topological ordered space (X,;:;: ,T; is said to be
decreasing generalized closed (written as decreasing g-closed) if
and U is decreasing open in X. Similarly increasing g-closed defined
;;;;U
3.2 Definition. A subset A of a topological ordered space < said to be
increasing generalized open (written as increasing g-open; if X-A is decreasing
g-closed. Similarly decreasing g-open defined Clearly increasing open (resp. decreasing closed 1 :::ets are inceasing g-open
{resp. decreasing g- closed) sets, but the converse is not necessarily true.
3.3 Definition. A topological ordered space ( X,;:;: , TJ is called a lower (upper) T 1
z-ordered if every decreasing (increasing) g-closed set is decreasing (increasing!
closed. (X,;:;: ,T) is said to be T v2-ordered iff (.X,;:;: ,Tl is lower and upper T 1,2 -
ordered.
3.4 Theorem. A subset A of a topological ordered space (X,;:;: ,T) is increasing
(decreasing) g-open iff F;;;;I0 (A)(D 0(A)) whenever F~A and Fis increasing
(decreasing) closed.
Proof. Necessary: Let A be an increasing g-open and suppose that F;;;;A \vhenever
Fis an increasing closed set. By definition, X-A is decreasing g-closed set. Also X
-A r:;;;_ X-F. Since X-F is decreasing open, therefore D(X-A);;;;; X-F. Hence X
J0(A) ;;;;X-F and so Fr:;;;_JD(A).
157
Sufficiency : If F is an increasing closed set with Fe;;_ !0(A) whenever Fe;;_ A, it
follmvs that X-A :;;;_ X-F and X-I0 iAJ :;;;_ X-F. Hence DCX-A) c;;_ X-F. Thus X-A is
decreasing g-closed and so A is increasing g-open. Similarly, the decreasing gopen case may be discussed.
3.5 Definition [l]. A topological ordered space (X, ~ ,T) is said to be lower
(upper) regularly ordered if for each decreasing (increasing) T- closed set F:;;;_X
and each element aEF, there exist disjoim T-neighbourhoods U of a and V of F
such that U is increasing idecreasingi and \ 1 is decreasing (increasing) in X .
< ·1 l is said to be-regularly ordered iff (X, ~ ,T) is both lower and upper
regularly ordered.
3.6 Definition. A topological ordered space (X, ~ ,T) is said to be lower (upper)
g-regularly ordered iff for each decreasing (increasing) g-closed set F c;;_ X and
each element a£: exist disjoint T-neighbourhoods U of a and V of F such
is increasing f decreasing1 and Vis decreasing (increasing) in X. (X, ~ .1 J 1s
said to g-regularly ordered rX, ~ ,T) is both lower and upper regularly
ordered. It follows from the above definition that every g-regularly ordered space is
regularly ordered space, but the converse need not be true. Also a space is :regulady
ordered and T 1 r ordered iff it is g-regularly orderd.
3.7 Example. Let X={a,b,c} equipped with the topology T= I and
\Yi th the partial order ~ · defined as : a~ a, b ::_ h, b ~ c, c ~ c, then (X, ~, T) is a g
regularly ordered space.
3.8 Theorem. For a topological ordered space (X, ~, T) the following are equivalent:
Xis lower (upper) g- regularly ordered. For each xEX and every increasing (decreasing) g-open set U containing x,
there exists increasing ; decreasing) open set V such that
x Ev c;;_ I(VXD(V)) :;;;_ [:.
Proof. (a)=> {b). Let U be an increasing (decreasing) g- open set containing x.
Then is decreasing (incresing) g-closed set such that x liE X -U. It follows
that there exists a decreasing lincreasing) open set Wand an increasing (decreasing)
open set V such that X-U c;;_ W, x EV and W n V =qi. Since W is decreasing
(increasing) open and v,,....,,lV=¢, therefore wn1(vXD(V))=¢. Thus XEVc;;_l(V)
158
(D(V)) c;;;._X-W ~ U.
(b)=> (a). Let F be a decreasing (increasing) g-closed set and XE.F. Then X-F is
increasing (decreasing) g-open set such that x E X-F. By (b), there exists an
increasing (decreasing) open set U such that XE Uc;;;._ f(U)(D(U!) ~X-F. Thus Fc;;;._X-
/(U)(D([J)) which is decreasing (increasing) open and U,~ (X-J(U)(D(
Xis g-regularly 0rdered space.
=$.Hence
3.9 Theorem. For a topological ordered space (X, ~, T), the following are
equivalent: (a) Xis lower (upper) g-regularly ordered. (b) For every decreasing (increasing} g-closed set F, the intersection of all decreasing
(increasing) closed decreasing (increasing) neighbourhoods of F is exactly F.
Proof. (a)=> (b ). Let F be a decreasing (increasing) g-dosed subset of X and x i;=F.
Then X-Fc;;;._ U is an increasing (decreasing) g-open set containingx, therefore there
exists an increasing (decreasing) open set G such that x::: G ~ J(G)(D(G));;;;; U. Hence
Fe;;_ X-1\G)(D(G)) c;;;._ X-G and XEX-G. Thus X-G is a decreasing (increasing) closed
decreasing (increasing) nieghbourhoods of F which does not contains x. Thus the
intersection of all decreasing (increasing) closed decreasing (increasing)
neighbourhoods of F exactly F.
(b)=> (a). Let F be decreasing (inc:reasing) g-dosed and :q::F. There exists
decreasing (increasing) closed decreasing (increasing) neighbourhood A of F such
that x E:A. A is decreasing (increasing) closed decreasing (increasing) neighbourhood
of F implies that there exists a decreasing !increasing! open set \l such that
F;;;;Vc;;;._A. Now, xEX-A which is increasing decreasing1 open and Fc;;;._V which is
decreasi::1g (increasing) open such that V-. X-A = ¢ .
3.10 Th•?nrem. A topological ordered space (X, ~ ,T) is g-regularly ordered iff for
every set A and every increasing (decreasing) g-open set B such that A.~. B:;::: qi ,
there exists an increasing (decreasing) open set G such that An G * <jJ, and
/(G)(D(G)) c;;;._ B.
Proof. Let Ac;;;._ X and B be an increasing (descreasing; g-open such that An B:;:: qi .
Let x EA n B. Then there exists an increasing (decreasing) open set G such that
xEGc;;;._l(G)(D(G))c;;;._B. Clearly, GnA*~ as xEG.-A and (G)(D{g))c;;;._B.
Conversely, let F be a decreasing (increasing) g-closed set and x E F. If A= {x} and
B=X-F, t;hen A11B-=$. There exists an increaisng (decreasing} open set G such
159
that Ar G;:: 4i and J(G)(D(G)) ~B. F=X-B ~X-l(G)(D(G)). Thus XEG, F~X-J(G)
(D(G)), G is an increasing (decreasing) open, X-ICG)(D(G)) is decreasing
(increasing) open and G.~·(X-I(G)(D(GI)= ~·
3.11 Theorem. A topological ordered space tX, ~ ,T) is g-regularly ordered iff for
every non-empty set A and any decreasing (increasing) g-closed set B satisfying
An B = 9, there exists disjoint open sets G and H such that An G * <j> and B ~ H,
where G is increasing (decreasing) open and His decreasing (increasing) open.
Proof. An B == ¢ ~ A.~ (X - B) = ¢. Therefore, there exists an increasing
{decreasing) open set G such that An G= ~ and l(G)(D(G)) ~X-B. Then G and
H =X-HG)(D(G)) satisfying the required property.
Conversely, F be decreasing(increasing) g-closed set and x ff. F. Put A= {x} and B=F Then there exists an increasing (decreasing) open set G and decreasing
fincreasing,i Qpen set H such that A;;; G, B ~Hand G n H = <j>. Hence Xis g regularly
ordered.
REFERENCES SJ). ].k Separation a.x:ioms for topological ordered spaces, Proc. Cambridge Phil. Soc. 64 (1968}, 965-973.
[2] B.:M. Munshi, Separation axioms, Acta Ciencia Indica, 12m. (2), 140 (1986), 140-144. [3] L. Nachbin, Topology and Order, Van Nostrand Mathematical studies 4, Princeton N.J. 1965. [4] A.R. Singal and S.P Arya, Some new separation a.x:ioms in topological ordered spaces, Math.
Student, 40A (1972), 231-238.
Jnanabha, Vol. 37, 2007 fDedicated to Honour Professor S.E Singh on his 701h Birthday)
ON THE DEGREE OF APPROXIMATION OF EULER'S MEfu~S By
Ashutosh Pathak and Namrata Choudhary School of Studies in Mathematics
Vikaram University, Ujjain- 456010, Madhya Pradesh, India
rRecei1.:ed : Nouember 15, 2007)
ABSTRACT In the present paper we obtain the degree of approximation using the Euler's
means of function belonging to the generalized Holder metric and two corollaries
are also obtained.
2000 Mathematics Subject Classification: Primary 41A25; Secondary 40G05.
Keywords : Generalized Holder metric, Banach space.
1. Introduction. Let C2:: be the space of all periodic functions f on [0, 2n]
>Vi th series
x ~
- ~(a,. coskx + b;. sink.."C) =""A (x) 2 .L... '· .. .L.. k k=l n=O
we defined the space H w by
H "-' = {fo:C2:: : - f(f ~ '.O: kwfjx - ... (1.2)
and the norm . by
= . !
) ~}') t ... (1.3)
where ),
and D.u.•f(x,y)= f(x)- f(y) . w·11 '.)· ,x = v 1X-)" ~ ,,, f
... (1.4)
w(t), w • (t) being increasing functions of t.
If '.O:.A)x- • (1 I) I ljj and w ~x-n '.O: h1x-y , ... (1.5)
0<a::::;1, 0::::: 13 <a A and k being positive constants, then the space
l • ( ) ( \ I Hn = 1{sC2-::. : if .x - f y 1 '.O: k'x - O<a:s:::l
162
is a Banach space [1] and the metric induced by the norm
Holder metric.
on H0
is said to be
Let E,;(f;x) be the Euler's (E,q)-means for q>O defined by
1 ll ( n E q - '\"'I
n 1- .. )n f-..\ k ~ q, R=O· .•
;x)
where
sk(f;x) = __!_ c f(x + t )sin(k + ll2)t 2n· 0 · sint.12
is the kt\ partial sum of Fourier Sereis (1.lJ.
we write
~)t)= f(x+t)+ f(x-t)-2f(x).
2. Previous Results. Mahapatra and Chandra [2] in the year 1983 obtained
the degree of approximation of Euler transform of the Fourier series of t in the
Holder metric.
Theorem A-Let 0:Sf3<a:S1. Then for f'sH,,
/IE~(f)- t/113
= o{n(a-f3)i2(lognfia},
where E,7(f;x) i::; the Euler mean of the Fourier series.
Then Chandra (3] gave a better result of above theorem A. in 1988.
Thorem B-Let 0:SJ3<a:S1 and let fdI.,. Then
IJE%(f)- tllJl = o{n(Jl-a)logn}. . .. (2.2)
The object of this paper is to obtain the degree of approximation on the generalised Holder metric, using the (E,q)(q>O)-mean.
3. Main Result. Theorem -Let f E H1c, 0<n:S1, 0 :S l3 < ri and let w(t) be
the modulus of continuity, then
q(f,x)-ft. = o[Iogn(w(n/n)Y-13 '1]. . .. (3.1)
For the proof of the theorem, we require the following lemmas : 4. Lemma.
(i) Qn(t)= O(n) O<t<rr/n •w•(4.
(ii) Qn (t)= o(r1) n/n < t < n ... (4.2)
163
Proof : It is given that
n ( n, Qn ')-n~I =(h-q L.., k
' k=O\
sin(h + 1 2 )t 2sint 2
... (4.3)
Using the estimates
<(k..:..1(·)\_; - "" ' f ....... Jl and sin(k + 1/2 Xt s 1
respectively, for the proof of lemma 4i.i! and Lemma 4(ii) with the fact that
'n-1 n { n (l+ql ') ' q~-' =l. 'k:{/, 1tz . ,
we get
n-a ;,.,.:...._ ... ) ~ '·kn-k(k + 1/2}t It ~:;::·D\ R)
r ll { n''\ . I (nX1 +qt" I! k p"-r.
= 0(1 ~ - - k=O\ . /
'l(r1X1+qt''I1 n l k;Q
I= (1 _;_ q ~ n '; n-k(l!t) / q \ t:o ,k
Hence,
Q,,(t)= O(n) and Q,,(t)= o(r1 ).
5. Proof of the Theorem. It is known that by (1.6}
a . 1 n' n n-k E,; (f;x)= (
1 )"I k q S1:(f;x)
J. - q' .':=0 v
and let
cr (x) = Eq (f· n, , n ' l=--- sin(k + 1/2 }tdt
Similarly
<Jn
and
CT n (x;
=Eq n
(x)-un
1 ' ('") ' Q ; n 1 \ . )' , I' n \ --- I in-k~·. sin t / 2 k- I k jq :sm(k + 1/2 )tdt
-0\ / 2n(l+q,f
= 1 :::9At)-9)t) n [nl - - ! n-k . 2rr(l+q)" ·O sint/2 1~ kjq sm(k+l/2)tdt
164
1 r i,, + c l<1> Jt )- cj> y (t ~ f. n n-k · .. · 1 i. . Slni f:z-,- - 1[ dr;
2n(l+q)" .b •.- 12 sint 2 k=Oi k \ 2
= 11 + f 2 (Say).
Now,
/<!>Jt)-<!>y(t~ = + t )-r- f(x-t )-2f(x)}-{f(y + t )- f(y-t
~ jf(x+t)- +if(x)· -f{x-t'i+lf(y+t)-f(y)''.+ ! - \ !; j "'"' . l
~ kw(t )+ kw(t )+ kw(t )+ kw(t)
:-::; 4kw(t)
l<1>Jt)-¢y(t~ ~ lf(x + t )- f(y + t~ +!f(x-t)- f(J -t) _c_ 2f(x)- fl v)
From (5.2)
~ kwijx + t - y-tl)+ kw~x-t- y + tj)-e-2ku'(x
~ 4kw~x - yJ).
11= 1 r1nJ$_,;(t)-$y(t~f(n' 2n(l + q )" o sint I 2 ;;;;;J k j
sin(k-1/2 )t
= 0(1) [rr/ n w(t ')Qn (t )dt .o
= O(n) f~-rin w(t)dt
= O[nw(n!nXn/n)]
= O(w(n!n)). Consider,
by (4.lJ
lz = 1 frr l<Px(t)-<!>y(t~ n (n'\ n-k .
2n(l+q)"··rrln sint/2 6l.k? ·sm(k+l/2)tdt
=0(1)f. w(tXl/t)dt :r.;n
= o[w(nl n)jlogt] :/ n
= O[logn w(n!n)].
by (4.2)
f(y-
a...nd
(5.
(5.2)
(5.5)
H
i--3
/::>'
i;:j en
Cl
._..,
;::i
II
~
a ~
'."--~-·
0 II
crq
H
(fJ
(:::1
i;:j
. ~: 'd
q :1
·' .,
'" I ·c.~!
r;::::
:;-•
0 ·-
~ crq
,_
, (:::1
r.<
.
'-; II ,..:·:
CJ\
-!
;';-·
/1
:.1
:1
;:::s
~~.
~'"
ij
·O
1:-,'.
) : ~ ..
~
n (K
l 0
s. s
~j
c::r'
......
,_,
i;:j
II 8.
(f
J .... ::;J
:::J
crq
crq
:1
§ 0..
::1
0
1
~M _
_ J
01
166
Thus, collecting the above estimates, \Ve have
llE%(f;x)-f(x~lw* = o[Iogn(w(n/n))1-i'
11]
This completes the proof.
Corollaries . If we put Tl = _ "i)< AixJl - ' ,w"(ix- .Y)s:;k1x-\ . .
Corollary 1. For fsH0
, 0 <a :S 1, 0:S13 < a for ali xs[02:c J then
jjE% (f; x )- f(x Ji = o(np-,, (log n )13• (1 ) O < a < 1
!I O\nfH(logn)li) a=l
Corollary 2. Let fs Lip a; O < a :s; 1 , then
llE%(f;x)-f(x~I = 0(1Xn-a logn)
Proof put 13=0 in the above corallary.
REFERENCES [1] P Chandra, On the generalized Fejer means in the metric of Holder space, Math, Nachr, 109
(1982), 39-45. [2] P Chandra and R.N. Mahapatra, Degree of approximation function in the Holder metric, Acta
Math. Hung.,AI (1983) 67-76. [3] P Chandra, Degree of function in the Holder metric, Jour. Indian Jlath, Soc. 53t19&8J 99-114. [4] S. Prossdorf, Zur Knovergenz der Fourierreihen Holer si:eriger F!.mktioneR Math .. Nachr. 69
(1975), 7-14. [5] T. Singh, Degree of approximation to functions in the Banach space. Mathematics Student, 58
(1991) 1-4, 219-227. [6] T. Singh, Bijendra Singh and Surjan Singh, Degree of approximation w a class of continuous
functions with Banach space, Vikaram Journal of }.fathemat£cs, 5 No.1 (2005), 289-296.
167
Jnanabha, Vol. 37, 2007 (Dedicated to Honour Professor S.P Singh on his 70'h Birthday)
MULTIVARIABLE ANALOGUES OF GENERALIZED TRUESDELL POLYNOMIALS
By R.C. Singh Chandel and K.P. Tiwari
Department of Mathematics D.V Postgraduate College, Orai-285001, Uttar Pradesh, India
E-mail : [email protected]
1Receiued : Nouember 20, 2006)
ABSTRACT In the present paper, we introduce a generalization of multivariable
analogues of generalized Truesdell polynomials, due to Chauhan ([5], p.112, (6.1.3). Our pol:y-nomia&s may also be regarded as multivariable analogues of generalized Tru.esdell poly""IHJmials due to Chandel ([1],[2],[3],[4]). In the last, we also introduce f1uther genedization of our polynomials. 2000 iiathematics Subjeet Classification : Pramary 33C70; Secondary 33C50. Keywords : ~'iultivariable Analogues, Generalized Turesdell polynomials, Generating relations, Recurrence relations, Rodrigues' formula.
1. Introduction. Singh (6] introduced Truesdell polynomials defined Rodrigues' formula :
1 .... (x.r.p )= x-'"e1"' (/'
. d l --x~ O=. dy"
, .....
Chandel [1},[2],[3],[4]) studied a class of polynomials defined by Rodrigues' formula :
(1.2) .r.p)= l ~ Q =
) _'i; -
d
dr'
where k :t:- l ·
Fo:r k--+ 1, 0.2) reduces to (1.1).
Recently, Chauhan ( [5] ,p.112, (6.1.3) introduced and studied a multivariable analogue of generlaized Truesdell polynomials defined by Rodrigues' formula
l.3J ,,,_ . ..r~;p, ..... p,,1(x1 , ... ,xm)
_r - . - .r,)_ -( - rm)·~b -Ll logx1 p 1x 1 7 ••. a"' logx,,, Pmxm j
n.i. - n= ~ - ( - r-1 \ - - - i~n . -b ' } 81 ... om [1 \a 1 logx1 p 1x1 ) ... (a"' logx111 PmXm )]
168
where n1
are positive integers,ai ,r;, p 1• b are arbitrary numbers real or
r complex independent of all variables x,; o. =x,-~-. i=l, ... Jn. ' . . Cx.' . '
It is clear that
( 1.4) lim T; 0
b~:.:: '~1··--.h,,,
b ·---.J~.,
l,. . .,xm)
=Ta, (x,r,p; ). n1
,rn: ;pm)
Motivated by (1.2) and 0.3), here in the present paper we introduce multivariable analogue of Chandel polynomials ([11,[2],
{r,;~:_'_''.1;·,:·"1 " J:, . .. ;:,,, ,., .... .J;,.:p, ..... p,,, 1(x1
, ... ,xn, )/ n, = L2 .... :i = L ... ,m
defined through Rodrigues' formula:
Cl.5) T(i ·--~1 • ... u.m :J:! , ... ):,"';rt, ... ~!', :pl, .. ,p"-n1 ···-n:,,,
,. ... x·) , n~ ,
= [1-(a 1 log x1 - P1 )- ... - log Xn - P~-< lJ
Q"1 ••• Q 11"' fl - (a 1 log x 1 - p 1 x~·, X; .1::11 l ' log
where n; are positive integers, b, a1
, k; ""1, ri, pi are
real or complex independent of all variables x. :fb.:_ = x.
For k; ~ 1 (i= 1, ... ,mJ, (1.5 reduces to .1.3).
From (1.5) and (1.2), it is also clear that
i . b (1.6) lim T
b---t'Z n1,---,1Zm
b .... ,h111;r1, .. ,r,,,.;bl --~-
; ..... .I:
-Ttu,.h,1(. ·) Ti",,.1' ... - n
1 X1,r1,P1 ... nn ,r_ry: ,Pn:)
:::
~-· t..-.".
num"bers
m.
where T,;"·k 1(x,r,p) are generalized Truesdell polynomials due rn Chandel
([1],[2],[3],[4]) defined through Rodrigues' formula il.2J. F--r brevity, we shall write
~;~_:1/~: .. ·'' ):, ... .J: .•. :r, . .r"'.D .,p,, l(xl ,. .. ,X,,,)
as
169
: , ... ,x,").
2. Generating Relation. Starting with Rodrigues' formula (1.5), >>:e have
.::dJ
\ t~!; t ~:-(. x )-... ' x, , ... ,. n; • nn·
cc
I:
r = ll- 1 logxi - p,x; 70 logx" - P.,:xn:
0
ex'P(t1Qx, + ... +tmQx,,,)
logx1 - P1 r log Xm - PmX;{ } Now applying the familiar result due to Chandel ([1, p.105 e:q. (5.2.5);
Also see Srivasta,:a-Singha1 [7,p.76 eq. (1.12)]
.1 i-
,. ·'
where \Ye have
fix r admits Taylor's series expansion,
o:
) ..:...- , ... ,x~) ~~1'! ... ~:,.· '.
=-0
f -ll- loa··· -Dr.· - ~ b~"~: .... ~-- logx -'] b
lJ
r, P1X1' 1- loa X-b .l.._
{1- {1-(k1 -l)t,x~'-1
'• ... -
1 X. - a og ,,,
m [1- (k . - l)t.,.xk"'-l ]1!::.~lj , rr~ ,. ! rr:
Pmx:;~·
[ ]
r,., k,,.-l (k -1) 1-{km-l)tn:Xm .'m
Therefore, 've finally derive e generating relation
n.
~fl; f lL.1 l · <.In~ ( x ) _l ••• - ·,
Xi,.··,- m nl1 nm: (2.2)
170
[1 ( 1 r, ) { loo- r ... ·11J = - .al ogxl - P1X1". - ... -\a,,, oxm - PmXm .
r ( 1 l ) a i l I (k ·)t ;,, -1 I _1- a 1 ogx1 + ... +am ogx 17, -'- (k
1
-l) og(l- 1 -1 1x 1 ' J
+ + -d r, l . !:.-' )- , . f"""p,x, 11-(h, -llt,x,· · 1• ::,-1
' - .:.. - , ~ .:..
+ ··· + Pmx~;· {1- (km - l}t,,,
3. Application of Generating Relation. ~faking an appeal to generating relation (2.2), we have
f- rtb:[u];[k];[r];[p]i(x ·) t~'' t,,, L.... n1····,nm l , ... ,Xm. ! ... l
nr1 ... ,nm=O n1 . nrr..
- U:· . t n, t'''m or. L T~~,:!_u};~,~];[r];[p]J(xl , ... ,xm )-1-! ···~ Y n1 , ... ,nm =0 n 1 " nm : S, .... S., =0
Therefore, we finally derive
(3.1) T(b+b';[a];[k];[r];[p]l(x1 , ... ,Xm) n1 .... ,nm -
= f ... I("' } .. (n,,, 5 1 5 m
Sl :::::Q Sm :=Q
which can be further generalized in the form
(3 2) T(b1+ +b";[u];[k];[r];[p]j(x ... X ) " 12 1 , .. ,ll.111 1' ' IH,
I I q , .
D(;.'. J 11,
S; 1 + ... +slt1=n1 Sn:· + ... +smu =nm }=l
T (b,1 ;[a ];[k ];[r j;[p ]) ( ) . - X1, ... ,Xm . s14 .... ,s...,'-l
(x t:' ts,,
1, ••• ,x ).-"- "' m S1! ... ·-i
.,. ! •. ~ ~ -·- ry;
l, .... X,_.
4. Recurrence Relations. By making an appeal to (2.2) we have
(l-cx 1 logx1 - •.• -am logxm + p1 x~1 + ... +Pm
z ) .:.......
(x ' t:' e~ . 1 , .... x )-' - ,,,, - , rn ••• -·-
S1 ! S 1 - !'P ~ I~ ::::G
n:
- t"' ('"' ( ) 1 m [1(·1 . . l ') x 1 , ••• ,x,,, --... -- - o: 1 ogx, ..,. ... .,-CJ.,, iog-x= l ! ' "' ~ ,,., ni. nm.
171
- a1 ~· kl~.
am ~ (k -1}'"' ... --- L- . "'r;;
='.:
-It'"' t,~~· + _ ... -'"'I" (r11(k -l)) s l P1-~1 i . Ill. s,
,,, =ll S 1 [
f i:, -I XI.
~' , _ ~ (7'rr,/(km-l)t ( ) ,. 'l l1' ""." ... - p.,.x_·- J •m k '-ls,,, xv'~-··sm
l i
~ ,---: sm! m ~m ! .
equating the coefficiants of t; ... t~'," both the sides; we derive
the recurrence relation
·' 1 h l - - - 0 l er )" - · --'- --'- p xr· I 4. \ -a.I .ogxl ... '-'·c: 00"!7: P1 , ... , m m /
= [1- (a. logx, + ... -;-a,., logx.,, l ... . ' ><-
(xl , ... ,xm)
,'1,
+ r/n- -CJ. i(h. -l)Vh -l)s,X(k,-l)s,T(b;[a];[kj;[r]:[p])(x ..... X ) 1 i .A 1 1 n;, -st ,n.2 _____ n
1:-: 1 ~ · m
='~"
~ -l)t -CJ.m /(km -l)Jk111 -1)8•• -1
cf =·"'
', ... ,x,,,, ,1
Differentiating (2.2) partially with respect to ti' we have
[1- logx1 - p 1 x? )- ... -(a.m logxm - Pmx';;, )]
:.>:
'\ ~
=0
=-0 '\ ~
n, .. .,21co=G
r 0:
-a, -l) =0
, t':C:-l r'n" r·"·· ) ' ·) m
..... x,,. :-1Y.n-2l ... nm!
t"' r )_l_
•••• , .. ..,1711 l n1·
t""' m
n ! m·
-l t + P1T1X~:~k1-l f t:1
s, =0
Tl ) f 1 kl -1+1 l(k1 - l)x~,-1 y I.
s, J
Thus equating the coefficients of t~1 ' ••• t;;" both the sides, we derive
the recurrence relation
{·4 2' [1- l p;~ - - 1 + J; + , r,..] T(b;[a];[k1(rk[pj)( ) .. ) Ur o>:>"'"l ... am ogxm , P1X1 ' ... --r PmXm n+l,n2,. . .,n,, X1,. .. ,xm
;:.
b f:.-1 ~ = a.,x,' - ;
J_ .... ~
- 1) h: -1 l_s, r ( n, , X; } S1. S; (xl , ... ,xm)
S; =U
172
-b 1r,x:-k-l ~ 1 '~· 1 _3__1· r(k, -1J'x~:-'J1sl 'P _ - I L \, : k -1 L - -
;)1 :;;;(J \ 1 s!
, ... ,x ) r.'
which suggests that m-recurrence relations can be expressed in the following unified form :
(4.3) [1-a 1 logx1 - .•. -am logxm + p1 X~' + ... + Pmx';,;']
l!,
b k.-1 "'\' f r: _i ls. = aix;· L..si_ ,
s, =0
nl i J r r+k-l ,/n, l --'+l
-bp,r1x/ ' L \s, ': k -1 =D \ z
• J
s.
)5. J
•.... n_~< , ... ,x·) r;: _ .
n 1x,, .. .,:X~)
-'"r": \ .,. u~
, .... x·} - n: ' ~
Differentiating (2.2) partially with respect to xi' we have
[l-a1 logx1 - ... -am logxm + p1 x~' + ... -'-Pm n:]
~ - ~ T,~~:.[u-~~:1(d(p])(x1 , ... ,xm) t;', ... t;,;·, "1·····'-,,,-0 1 nl. nn:.
- ~ ~~ ' . ) ~ - b(-a1/x1 · P1r1X1 L..
n 1 .... .rr~; =0
... ~w~~~~""""':
+b i T(b+l;[a];[k];[r];[p]}{ _ )t;1 t~;- O:l . , n,,. .. JJ,,,=l' n1,. .. ,n,,, X1, .•. ,X,,, n,1·-· n_.,._: -7t1. X-
-1
f [(k1 -l)x~'-1 Y' t:' - P1X~:-k1-2r1 i h kl +lls, S1 ! s1 =0 s1 =G
n'T:'.
) ' 1 ls· -1 x~,-- j' tt' .
Now equating the coefficients of both the sides, we establish
(4 4 ) [1 log 1 g , _r, . , r ... 'JI . -a1 x1 - ... -am o x,,,,p1:k(..,.. ___ -,-pmx~,·
o T(b:[a];[k];[r];[p]l(x1,. .. ,xm)
~,.,_,... n1,···/l-n U-'-l
a1 , .r1-1JT(b;[a_]:[k];[ri:[p])( ) . b_- a1 --_-, P1T1X1 n1,. .. ,n"' X1, ... ,xm . ..,.. : -_-
X1 , Xl , .. .,xm)
-u_. -1 rz.
-2 "'\ .::.....
_•)
-r.
-:::{;
-1 ,y n1!
l . ' )' 0 (n1 -1- S 1 ;
- 1 l "'\ I ., -: J r--'~c>' s, h: -1
-l)x~'-i Jr
, ... ,x ) 1n
173
1
,,,.,Xm) !, _j
suggests on-recurrence relations in the following unified
{5.1
fl -,i
' c l ' --'· ' -'- p '(::~ ) -~-'OCT 1: -,- p-.t,· < --· ' nl" m , rev. .i ;e- j?~ ,;. .._ t....-·Ai
=b'-C!. -:c
L... -,- <"" -1 '~ -!_ i
l
r , ... ,xm )+bi ai
L X;
-·-· " ll~ c:-c] r; · 1 1 f(k l)xk,-iy - _k C s/~(1 ~• ' k,. ~ ; (' "i - , i I
,--.,Xe)]
5. Generalization. Consider
X- ~~-·~:::---,.
'l -1 - - l 1· - r ·•· · - - r• l v · - ·'•· 1 . .- -!-.v ...... J."".-Ol:::>X 71 Prr.XTr.J
Q_~; ... Q~= O' y .r;" 'l'· - - ... +an; lob '"m - ,vmxn; ,} ,
where
U5.2l G(z) = i> =0. n=O
For , (5.1) reduces to (1.5)
-1 ?!: - ..J... 5.11 defines
(5.3) ..... x~)
J, p). . .r;~'- j;_ r p ) : ll~ ' ,'71 . :
, ... ,xni.)
, ... ,xm)
(xl,. .. ,xn)
i=l, ... ,m.
174
where T,~aJd(x,r,p) are polynomials due to Chandel [l],[2], [3],[ 4]) defined
by ( 1.2) 6. Generating Relation. Starting with R?drigues' formula (5. lJ,
we have
f G~[,a]:[·~'!dr];[pj)(xl , ... ,X,,.) t;= ... t~,· n
1 ••.• .11
111:;;::U . " n~~ nm!
(6.1)
=[Gla1 logx1 -p1x~ -c- ... .,,.a,,,logx.n -p.,,x.~; I n • o )• (. r " \j
1--x,-·--',,,--.,,,, 'r!.a.logx -p.·c.'+ +a loax __ -p x··-·)' e C'-' \ J: l i- l .... < m b .111 m- m ,)
= [c(al log X1 - P1 x~' + ... +a,,, log xm - p"':<;
( 11 (1',-1) allogl rl-(hi - ,\, k,-1 - -\ l
( 11-
-1 h U:,-1)
.\
Pm<;;· r
/ Xm
lo · xk,..-l -'-- .. +am g fl (k -l)_tm m , . ~\ - m L (1 . . .. . . ·11- fl - l)t x"--·-1 ·i;;:,,,-ii
- !Ti. l72~ m
Thus we derive the generating relation
~ G([a];[k];[r];[pj){ ) tt t;> L ni, ... ,Jzm X1, ... ,Xm ,··· t n, , ... ,n..,, =0 nl; nm;
= [c(al log X1 - P1 x;·, + ... +am log x,,, -
r ( X;
GI a, logl ~ -(k, - l)t,xt·-• P1X~'
~ - (k. - l)t. x!'1 -1 f' (i:,-u
' - l .!. }
f ~-
0 i .·"m k,,.-l ·[1 \.k_,-+ ... +am l gl h-(k -l)tmx,,; ' '\ t rrz -
Pmx;:.;
fi-(k -l)t xk,,-i ' l m . mm
7. Recurrence Relations. Differenting (6.1), partially respect to ti, we have
(7.1) ~ ·f1!'flfj [n' tn·- t"-1 n r: L., G~1aJ;tkJ:trf1P l(x .... X )_L_ ;..'1' ;·· - (~i' t;;~
, ,;l, ... ,n1:: 1- ' ni ! ... J .- ·~f r ••• --n1 .... ,'2~=0 n,: n- l' In -1 n ' .>. l- \ l • :-1 ·
·= (G(a 1 logx1 - p1 x~1 + ... +am logxm - Pmx~;' II · G'
175
a.,x~-1 - .. r,+k,-1 1-(k -1)t.(-=!L_1))'J k-1 p,rzx, ' ,, k -1
, ;X; \ i
= log -p r, _,_ -'-a 'ogx - rm )}-1 G' Xi 1X1 ' ··· · m 1 m PmXm J ·
"" i-.-lI a,x;· .. - l)x,k,-i
( ri 1) 1--+ "" k-1 r - r_.k_.xT"'"k,-1 'I\\,_ l S; {(k -l)xk;-l}Si ts;
... ""4 ~ L r z , i
s,=D Si. s,=-0
Similarly differentiating (6.1) partially with respect to ti, we have
"' • fn1 tn1-r t~rl tnr1 t""' (7.2) )
i:..- J_l_ ~ J 1~1 ...1E:._ .• Xm ··· ( ) •·• · · n1! nj_1!nj -1 nj+l! nm! r;.,~ ~ - .Ti-~7. ~\}
l g - '} ...:... -'- r 1 - rm )}-1 G' ~O I1 P:X1 ...... am ogxm PmXm ~ .
. r r )
"_rl ~wk -1L~;-l f1 ts) - fj.,.kj-1 ~ I.. ¢i +
1
Sj frk -1\..krl ~j tsi ajxJ L.. ~ j J-'-j J j rjpjxj L.. s 1 ~ j Fi J i
s1 =D si=O j•
Now eleminating G' from (7.1) and (7.2), we obtain
.L-1 ~ftk a. JX/ .L..,[l j -
si=IJI
r \ l -
1-+ll , < - • ,,., kj-1 Jo K r· P, ,.- . '.~.•.-1 I -1 k 1)xk.-l JtS· I .;; . -T p xi ' . - .' J J .r ; ; ; sl J J J
si=D j·
r I . t_ 0 11ai1k};lr};[pj)(x ) t~'. t,~1, t;,-1
t;i;.11 t;m l i ,,,_ n, •... J•m l, .. .,Xm 1 ••• , { ) --••• -
1-
' ", •.. .n.,=D n,. n, 1: n, -1 ! n. ! n I < ,_ ' • i+ l · m · J
"' k-1,......, = a.ix,· L s,=0
( 7i 1) -· -+ "' k-1
- l)xk;-l 'Js, tfi - rp-x"+k;-l 'I\ 1 s, 5(k - l)xk;-l }s; ts; zJio ~~z ,i_; f ~L ~ l
S;=O Si.
J
176
[
'1> tn tn,_1 n -1 n . "1
"". aUa1MHfPll(x x )L /-1 t/ tj~i' t:;,s 1.
L., "1····l'm 1,. .. , m ! ··· I ( J ... --1 n1····Jlm=O n_, n. i·ll· -1 n. I n I · .. l ;- J ;+l · m ·
Hence equating the coefficients of t~1 •• .t;:,m both the sides, we derive
m(m-1) recurrence relations in the following unified form :
kr1rrk _ 1\ .... kr11'1 n) a(Mfk1[r1fpJl ( ) (7.4) a jXj l~ j }Xj j (n. -S .) n1, ... ,n,_1,n;+l,n;+1• ... ,nJ-1•nrsJ,n;.1• ... ,n,,, X1,. .. ,Xm
' j J
( r.
ri+krl _;_ + 1 -ripixi k-1
J
[(kJ -1)x~r1 J' (;:)
G~~~-~-[~;!;~~t;.1 •... ,n1_1,nrsJ,ni•'•···.nrn (4, ... ,xm)
;,,-1St ) k-1}s, nil G([a~MHtPD ( x ) = a;X; '~A -1 X;' (n s \! n1····Jli-1,n;-s,,n,_, ... A,._,,n;+l,nj+l•• .. ,nm Xi, ... , m
i - ii-
-r,p,xi·•-·( k;r~ 1
+1 J.. Kk; -1)x~ , r· (:: l G~~-~~~!;.~~~,n;.1 , ... ,ni-l,ni+l,ni•l•· .. ,nm (xl, ... ,xm) iJ= i:;;: j.
REFERENCES UJ R.C .. S. Chandel, Properties of Higher Transcendental Functions and Their .l;pplicati.on.s,
Ph.D. Thesis, Vikram University, Ujjain, 1970. [2) R.C.S. Chandel, A new class of polynomials, Indian J. Math., 15 (1973), 41-49.
[3) R.C.S. Chandel, A further note on the polynomials
(1974), 39-48.
r,p), Indian J. Math., 16
[ 4] R.C.S. Chandel, Generalized Stirling numbers and polynomials, Publ. Det. Institut Mathmatique, 22 (36) (1977), 43-48.
[5] S.S. Chauhan, Special Functins and Their Applications in Mathematical Physics, Ph.D. Thesis, Bundelkhand University Jhansi, 2008.
[6] R.P. Singh, On generalized Truesdell polynomails, Riv. Mat. Univ. Parma (2). 8 (1967), 345-353.
[7] H.M.-Srivastava and J.P. Singhal, A class ofpolynomails defined by generalized Rodrigues' formula, Ann. Mat. P.ura Appl. (4) 90 (1971), 75-85.
Jnanabha, Vol. 37, 2001 (Dedicated to Honour Professor S.P Singh on his 70lh Birthday)
SP-QUADTREE : AN APPROACH OF DATA STRUCTURING FOR PARALLELIZATION OF SPATIAL DATA
By N.Badal
Department of Computer Science Engineering,
177
Kam.la Nehru Institute of Technology, (KNIT) Sultanpur-228118, UP., India Email : [email protected]
Krishna kant Department of Computer Science Engineering, Motilal Nehru National
Institute of Technology, MNNIT, Allahabad-211002, UP., India Emial : [email protected]
and G.K. Sharma
Department of Computer Science & Engineering, Indian Institute of Information Technology & ivfanagement, IIITM, Gwalior-4 7 4005, M.P., India
Emaial : gksharma [email protected] {Received: September 10, 2007)
ABSTRACT Emerging spatial database applications will require this availability of
various index structures due to the heterogeneous collection of data types they deal with. Modern GIS can accept different kinds of data after resturcturing and partitioning the spatial data depending on application. This paper introduces the new domain decomposition algorithm to create a valuable set of spatial data to speed up the performance of hybrid GIS Data set. SP-quadtree is an interesting choice for spatial databases, GIS, and other modern database systems. In the following section concerned algorithm is examined and the experimental results have also been presented in this paper. 2000 Mathematics Subject Classification : Primary 68P05; Secondary 68P15. Keywords: GIS, Quadtree, B+tree, hybrid data
L Introduction. Complexities are involved in the access of spatial data. There are problems in handling of these spatial database in- query operations in single machine and multi machine environment. A single query locks the entire spatial data to maintain data consistency for its sole processing and prohibits its access by other query process. Number of queries can run concurrently if the locks can be arranged in sub domain related to that query. There exists a possibility to partition the spatial domin data set into disjoint sub domain data sets. The data
178
structure in hierarchical fashion makes a provision for managing locks on sub domain. Thus a .query process will lock only the relevant sub domain of spatial database leaving other sub domain to be managed by other concurrent query on several computing units.
Therefore, a detailed study is required for the common features among the members of the spatial space partitioning tecniques aiming at the capability of representing and producing the partitioned spatial domain data. Moreover, a new extensible index partitioning tecnique is required to support the class of space partitioning data structures for a domain.
2. Space-Partitioning Trees : Overview, Challenges A single query locks the entire spatial data to maintain data consistency
for its sole processing and prohibits its access by other query process. Number of queries can run concurrently if the locks can be arranged in sub domin related to
that query. There exists a possibility to partition the spatial dornin data set into disjoint sub domin data sets. The data structure in hierarchical fashion makes a provision for managing locks on sub domin. Thus,a query process will lock only the relevant sub domin of spatial database leaving other sub dornin to be managed by other concurrent query on several computing units.
Emerging GIS spatial data applications require the use of new indexing structure beyond B+tree, R tree [01] [08]. The new applications need different structure to suit the big variety of spatial data e.g. multidimensfonal data. The space partitiong trees are well suited for such multidimensional relational spatioal data. The term space-partitioning refers to the class of hierarchical data structure that recursively decomposes a certain space into disjoint parttions. The structural and behavioral similarities among many spatial trees create the class of spaceparti tioning trees [03]. Space-partitioing trees can be differentiating on the structural differences (i.e. types of data, resolution, structure etc.) and behavioral differences (i.e. the decomposition principles). The main characteristic of spacepartitioning trees for spatial data is that they partition the multidimensional space into disjoint (non-overlapping) regions. Partitioning can be either spacedriven that decomposes the space into equal-sized partitions regardless of the data distribution, its means the space is partition physically [07], (2) data-driven that splits the data s~t into equal portions based on some criteria, e.g., based on one of the dimer1sions. This refers the partition is made logically for spatial data of space [07]. So if the principle of decomposing on the input data, it is called data driven decomposition, while if it is dependent solely on the space, it is called space driven decomposition.
There are many types of trees i.e. Height Balancing Tree, R-trees, SR-trees.
179
KD-trees, PMR-trees etc. in the class of space partitioning trees that differ from each other in various ways. Some of the important variations in the context of the tree data structure are: *Path Shrinking. Two different types of nodes exist in any space-partitioning tree, namely Indexed node (internal nodes) and data nodes (leaf nodes) [06]. The problem is associated as how to avoid lengthy and skinny paths from a root to a leaf. Paths of one child can be collapsed into one node. The address nodes are merged together to eliminate all single child internal nodes. *Node Shrinking. The problem is that with space-driven partitions, some partitions may end up being empty. In this strategy no indexed node will have one leaf node as user decomposes only when there is no room for newly inserted data items. * Clustering. This is one of the most serious issues when addressing disk-based space-partitioning trees. ·The problem is that tree nodes do not map directly to disk pages. In fact, tree nodes are usually much smaller than disk pages. Node clustering means choosing the group of nodes that will reside together in the sanie disk pages.
The quadtree structures, used to code spatial data, have been geometrically varied. A quadtree is a representation of a regular partitioning of space where regions are split recursively untill there is a constant amount of information contained in them. Each quadtree block (also referred to as a cell) covers a portion of space. Quadtree blocks may be further divided into sub-bloks of equal size recursively. In many hierarchical partitioning of the space has been in terms of regular-shaped squares and each may be produced to different users in parallel manner. However, with some structures [06] the data themselves have been used to determine the position and shape of the subdivisions. The distribution or position of the points, lines, and vectors may be used to divide the space into irregular shape at each subdivi.sion so helping to represent the variations in density and distribution of data values at various scales. The main problem with existing quadtrees with spatial data is the loss of explicit topological referencing [04], because of paths of sub-block can be collapsed into the block and that the precision of point and linear features are limited. So the existing quadtree struc~~re is not well suited due to the importance of referencing for hybrid relational GIS Data. The new proposed SP-Quadtree data structuring techniques may be used with hybrid relational GIS data where the presence or absence of an attribute determines the coding of a quadrant for parallel GIS and does not loss the topological referencing.
Another problems of node shrinking in which empty node exists after some
180
time this becomes valuable (e.g.flood, new building) and block becomes important. This work includes the concept of data warehousing i.e. non-volatile in SPQuadtrees and contains the previous information.
In comparisons, updating the attribute data is trivial; provide the one-toone links between attribute data records and the spatial entities remain unaltered. Traditi01ial database systems empoly indexes on alphanumeric data, usually based on the B-tree [01] [02], to facilitate effcient query handling. Typically, the database system allows the users to designate which attributes (data fields) need to be indexed. However,advanced query optimizers [14] [12] also have the ability to create indexes on un-indexed data of temporary results (i.e., results from a part of the query) as needed. In order to be worthwhile, the index creation process must not be much time-consuming as otherwise the operation could be executed more efficiently without an index. Spatial indexes such as the SP-quadtree are important in spatial databases for efficient execution of queries involving spatial constraints, espectialiy when the queries involve spatial joins. This research investigates the issue of ~peeding up building domain based on SP-Quadtree for a set of spatial
data and develops an algorithm to achieve this goal. Spatial indexes are designed to facilitate spatial database operations that involve retrieval on the basis of the values of spatial attributes. If the database is static, user can afford to spend more time on building the index as the index creation time can be amortized over the number of queries made on the indexed data. However, this research considers the case that the database is dynamic. This situation arises when the output of an operation. In this case, evaluation of the effeciency of appropriateness of a particular spatial index must also take into account the time needed to build an index on the results of' the operation. The time to bulid the spatial index plays an important role in fo.e overel!_p~rform~ce of hybrid relational GIS database.
N a-j--w s
I rOu -/7 l...rv1:l0
l NE NW
"CPI,, U-.-.11
S£
/I Y I /1 Y cyn,. ~12
Figure I: A SP-Quad tree Implementation
181
SP-Quadtrees can be implemented in many different ways. A key aspect of implemenation of the SP-Quadtree is its splitting rule in which regions are split recursively into quadrants. The method presented here, inspired by viewing them as trees, as shown in figure 1, in which SP-quadtree is two dimensional space in a 4-way branching tree that represents a recursive decomposition of space wherein at each level a square subspace is divided into four equal size squares (i.e. quad block or cell) labeled the north-east, north-west, south-east, and south-west quadrants. So the area of each quadrant at level 1 is one fouth of area of level 0. Area L 1=lf4 Area L 0 At level 2 Area L 2 = (1/4)2 Area L0 .
In general, the area or space at any level of one qualrant can be represent as Area Li = (1/4)i Area L0.
Quad blocks are managed in row, column order in SP-quadtree. At first level {n=O) SP-qu.adtree contains one row (i=l) and one column (i=l), at second level (n= SP-quadtree contains two rows (i= 1,2) and two columns (j= 1,2), at third level =2) SP-quadtree contains four rows (i= 1,2,3,4), and four columns {i=l,.2,3,4). at fourth level (n=3) SP-quadtree contains eight rows (i=l,2, ... ,8) and eightoolu:mns (j=l,2, ... 8). In general at a level (n=n), the SP-quadtree contains 2n rows (i.e. 1,2, ... ,2n) and 2n columns (i.e. 1,2, ... 2n). Each quad block contains the control point (CP) at central location for maintaining the reference of next level control points and denoted as CP Level row column inforior (e.g. CP 112 ). At level zero the control point is noted as CP011, at level one (n=l) SP-quadtree contains a set of four control points (i.e. CP 111, CP 112, CP121 , CP122 ). So at any level (n=n) there exist a set of control points, that may be represented as
{CPni)where l::;;i::;;2" and 1::;;j::;;2n.
Any control point at any level contains the referencing information of next level control points. For example at level zero, control point (CP 011) contains the information of level 1 control points as a set of four control points (i.e CP 111 ,
CP112, CP121 , CP122). The referencing set of control points at any level can be represented as
CPnij= {CPn+ll2i-l,2j-ll• CPn+l(Zi-l,Zi)' CPn+lczi,Zi-D• CPn+l<zi,2il}. The location and size with control points of each child leaf block is encoded
in such manner so that they do not loose the referencing and are used as a key into an auxiliary disk-based data structure. This approach is termed a linear SPQuadtree. If control point varied from original location to another point location is termed as Point SP-Quadtree. This situation is useful when space partitioning is done on the subject oriented bases.
182
3. Algorithm. This section gives the algorithm with description for spatial data domain decomposition. The SP-quadtree node definition in our spatial domain decomposition algorithm is described as the C language structures. This structures starts with initiation for length and level of the domain and decides the control points. When the domain is decomposed, it forms rows and column so this structure contains the information for start and end for rows and columns. typedef struct SP quadnode _ struct {
int length.;/* square side length of the node* I int level;/* a parameter to control how deep a node should be divided* I int control_pt num;/* number of control pants in this node*/ int startrow;/* start row number of this node in global grid*/ int startcol;/* start column number of this node in global grid*/ int endrow;/* end row number of this node in global grid* I int endcol;/* end column number of this node in global grid*/ } SP Quad.node;
For the implementation, a SP-Quadtree is stored in a single direction list as the following C ·language structure shows. typedef struct SP_ quadnode Jist _
struct
{ . SP Quadnode quad; struct object_list struct*next,/* Pointers to next SP _Quad.node object*/ } SP_ Quadnode list node;
Proposed SP-quadtree-based spatial domain decomposition algorithm is designed for general use and it can produce scalable geographical workloads that can be allocated to available computatio:nal resources and speedup the performance of hybrid relational databases. The algorithm addresses the decomposition challenges [05) that are centered on finding efficient data parititions that are assigned to each GIS resource. The SP _Quad_Decompose algorithm has two input parameters: level and min _length. In SP_ Quad_ Decompose, these two parameters togetl:ier with information about the number of control points at each SP-quadt:ree node are used to determine the level and location of recursive division. In the execution of SP _Quad_Decompose, those parts of a region that contain a high density of control points are recursively divided until the level of their SP-quadtree nodes reaches zero from a user specified value (greater than 0). However, regions with a low density of control points will not stop being divided until the square length of the node is less than the parameter min_length. Regions With an intermediate density of control points will invoke either of these two rules depending on which is satisfied first.
183
First of all the algorithm initializes the first node of SP-Quadtree rather than assigning the level to current node in the step 1and2. In the step 3, algorithm tests the conditions for splitting the node. If qual level is not greater than zero, stop the algorithm (i.e. steps 3.a.2). The algorithm is also break of quad length is not greater than minimum length through step 3.b. If current node has to be split, generate the four children for north-east, north-west, south-east and south-west regions of current quad node and assign the control points for them than current quad node has been removed. The four new children of the deleted quad node have been appended at the end of list. The algorithm is desctibed as follows in figure 2. SP _Quad_Decompose (int level, int min_length) 1. Initialize the first node of a SP _quadtree/* curr is the pointer that is always pointing to the currently accessed node* I 2. curr-> SP _quad. level =level 3. while (TRlJE) {
3(a). if (curr->SP _quad.level>O) {
3(a.1). if (curr->SP _quad.length>min_length) {
3(a.1.1). Intialize four children of this node: p_sw, p_se, p_nw, p_ne 3(a.l.2). Assign control points to four children 3(a.l.3). if (curr->SP _quad.control_pt_num>=SEARCHNUM) /*SEARCHNm1 is the k in k-nearest neighbor search* I {
p_sw->SP _quadJevel=curr->SP _quad.level-1; p_se->SP _quad.level=curr->SP _quad.level-1; p_nw->SP _quad.level=curr->SP _quad.level-1; p_ne-> SP _quad.level=curr->SP _quad.level-1; }
3(a.1.4). else {
p_sw->SP _quad.level=curr->SP _quad.level; p_se->SP _quad.level=curr->SP _quad.level; p_nw->SP _quad.level=curr->SP _quad.level; p_ne-> SP _quad.level=curr->SP _quad.level;. }
3(a.l.5). Add four children to the list in Morton order and delete the current node
184
that curr is pointing to control point. }
3(a.2). else if ((curr->next)=NULL) break; else curr=curr->next; }
3(b). else if((curr->next)= NULL) break; else curr=curr->next; }
Figure 2: An Algorithm for SP-Quadtree based Domain Decomposition The splitting module of step 3 of algorithm in pictorial manner is shown
in figure 3. This points the current node which has to split than generate the four children. The current node is removed and four new children are appended at the end of list.
List he>d
·= thu is b~'
Li<'. w.l
l l l ' I ' . i . . I I I
L ·- I I f ··- ---· ·~·· ,_ 1 =c . I , i .... ~ . , =-· -- .::.=::. ~ ....,, ~ .riiz...
lh .. .d I , ,tall , ! ! f J D~q:.;u..~hub~t::~..,...,...<t>Cdar!LJ": I : i I I ~th.1.V>e ~~e..~.e~d l j
i I r-ri
I~~ i
I ---r j s~v : SE ; }..""'k.. l }..'E l } I
y~-n~-~CF~Of ~-----=d!~
~dz: hos 1>•"" app~d "' th< C\d. crftl:'w ll£t.
i I
,~
L . ·--'----···-~-. __ J ____ l ____ ~------'-----~---'"---~..C Fi.gure 3: fonnation ry Quad ti\ing list in SP-Quudtree Al:;on!hm
Ut>ing two parameters: level and min_length, the SP _Quad_Decompose algorithm produces balanced workloads of two types: extensive un-interpolated areas with large number of control points and compact un-interpolated areas with small (quite often, equal to zero)number of control points. However, users can specify these two parameters to determine the granularity of workloads according to the computing capacity of their accessible GIS resources. For example, users can increase min _length from 65 to 125 to generate larger workloads. These values,
185
which are specific to this 2000 by 2000 problem, represent the side length of Hybrid cells generated from the Clarke algorithm [12]. These values can easily be converted to real world units or ratios based on the size of entire GIS data sets.
4. Performance Evaluation. The domain decomposition algorithms descrived in the previous sections are evaluated using the four datasets on SUN FIRE X4200 m2 system of two 2.4 Ghz duel core AMD optern 2000-series processors with 4 GB DDR2/667 ECG registered memory. The system architecture contains three 8.0 GB/sec hyper transparent links per processor and 10. 7 GB/sec access between each processors and memory, and has four channel SAS interface with SUJ>l RAY server softwere with solaris 64 bit. For each dataset, two parameters in SP _Quad Decompose algorithm (level and length) were adjusted to determine the parameter configurations that yield the best performance (i.e., minimum computing time), and also calculated speedup S, which is defined as S=tlitn(eq where tl is the run time realized on a single processor that is selected by the broker selected processor is the fastest one among all avilable hardware environment); and m is the computing time when multiple processors are used to conduct computations.
For the uniformly distributed data, the single parameter level was adjusted because length does not affect the process of decomposing the spatial domain. Different level values were chosen to observe which level value achieves the best performance. All resources are used except that when level is equal to 1, two sites are used because only four jobs are generated by the SP_ Quad _Decompose algorithm. It was found that when level is equal. to 2,the best performance is achieved because of the characteristics of the Grid computing environment and the problem being addre&.."8ci. In particular, there is an overhead penalty imposed on each job that is generated by the SP_ Quad _Decompose algorithm. This penalty can be expressed in the following equation: P=Ti+Tj(eq no.2) where Ti is the job initiation time.
Table A and figure 4 show that without decomposition it takes more time when decomposition starts on different level and different length. It reduces the computing time when at large level and length it becomes steady due to increase of decomposition overhead.
w·1u out I
Decomposition With Decomposition
l Level =2 ! Level= 3 Level= 4 Length 1 Length ! Length Length Length Len9th = 64 = 12S i = 54 = 128 = 64 = 128
Computing I 34.47 l !
Time (Sec.) 5G.4B 4208 l 43.16 55.17 4734 5-958
Table A: Comparison of Computing Time for With and Without Decomposition
186
70 r·- ---
I 60
~ 50
¥ 40 >= '~ l "'---ff;.\.~ laCanputkg 1im<> (Sec)~ :=" 30 a. §w
10
0 l f. I ! 0 ~~9 ! £ tt ) !r""''""'-"'J t · A l - I 1
~ - )
· I 1~~1~1~~1~~~1~~ I r 1 • 64 1 = 129 1 = 64 1 • 12s = 1;4 1 = 12s
i 1 j !.~:".;el: 2 ~ ~ = 3 La;.,e1 c 4 I I Wthout j Wth 0.ccmpo.,;"2on o.l:>orrcositlcn
Figure 4: Comparison of computing time
r
Figure 5 compares the execution time when SP-quadtrees for the point data with that for a general quadree. The execuation time appears to grow linearly with the dimension for both algorithms. This is to be expected, since the size of the point data as well as the time to compute geometric operations grows linearly with the dimension. The quadtree algorithm is slightly faster for all dimensions but the difference between the two techniques gradually decreases as the number of dimensions increase. This difference corresponds to the overhead (in terms of execution time) in the SP-quadtree algorihm due to the use of the pointer-based quadtree. However, the relative parity of the two algorithms is orJy achieved when the improved SP-quadtree algorithm is used. Without it, the execution time for the SP-quadtree algorithm grows exponentially v.'ith the dimension.
I 4(l ·--- - H -··-·--·---... -- -~. --~ I . I 3~ I ~ • I '.)0 .... .-- - I
K 25 ....- ~ . I E __ .,,. t .. , i" /< - 1 · ~=-~ . ~ 15 :;:;~~· ...,,~' ~',,.,,.~·~~~~-~~,~~~~.-:~:~.,..,..~~~~7"~~ .......... ~~~-l
' 10 ..... ....,,...;,,,..,._,....,..~ ..... ~....,,~ ...... ~....,.....-.,,.....,,~~~~~~~~~~
5 +-~~~~~~--~~~~_,,.. ............ ...,....,.~~~~~~~...,.
Figure 5: L~ecution Time for Quadtrees and SP-Quadtreesfor Point Data Sets of Dimensionality.
5. Conclusion. The space partitioning tree methodology is implemented in this paper. This makes it possible to have single tree index implementation to cover various types of trees that suit different applications. The general objective
187
of this paper is to investigate and speedup the performance of a hybrid GJS data set in parallel manner. A SP-Quadtree-based domain decomposition algorithm was developed and evaluated when used uniformly distributed and clustered datasets in the hybrid computing environment. The results show that for a dataset with a uniform random distribution,the domain decomposition algorithm scales well given the available GIS resources. More specifically, speedup is increased when additional re..«ources are used. As expected, sites with larger computing capacities contributed more to observed increases in speedup. Future work may include the same methodology for more complex oprations such as spatial joins.
6. Future Scope. To improve times related to a distributed query, the final cost of the operation shoud be reduced. This cost is made of processing and communication costs. The latter one have greater impact on the final response time. Despite the reduction of the number of messages exchanged between servers, it is important to emphsize the need for an adequate operating system and network tuing when using high performance network paths [09] [10] [11] [13]. Load balancing is another big challenge to be highlighted since geographic slices produced by the broker though hot considered here but may effect the space partitioning. The work may be extended to investigate whether the buffering strategies for bulk-loading may be used to speed up dynamic insertions and queries. Also, the fact that the system can build quadtrees eff ciently will enable user to build a spatial query processor that exploits this to construct spatial indexes for temporary results or for un-indexed spatial relationship prior to spatial operations on them.
REFERENCES [l] R. Bayer, The universal B-tree for multidimensional indexing: General concepts, Lecture Notes,
Computer Science, 1274, 1997. [2] J.L. Bentley, Multidimensional binary search trees used for associative searching, Compmunications
oftMAC.M., 190975) ,509-517.
[3] K Cbakrabarti and S. Mehrotra, The Hybrid tree: An index structure for high dimensional feature spaces, Proc. ICDE Sy'dney; Australia, (March 1999), 134-140.
[ 4] M. Egenhofer and R Franzosa Point-set Topoiogical Spatial Relations, Int'lJoumal of Geographical Information Systems, 5(2)(1991), 161-174.
[5} C. Faloutsos, H.V Jagadish and Y. Manolopoulos, Analysis of the n-dimensional quadtree decomposition for arbitrary hyperectangles, Proc. TKDE, 9,no. 3(1997), 373-383.
[6] RA Finkel and J.L. Bentley, Quad trees: a data structure for retrieval on composite key, Proc Acta Information, 4(1)(1974)1-9.
Vclgantini; An effective way to represent quadtrees, Communications ACM, 25(12)(1982), 905-910.
[8] Guttman, R-trees: a dynamic index structure for spatial searching in ACM SIGMOD, June(1984), 47-57.
[9] S.L. Lau, Disk, Network and Operating System Effects on GIS Performance, GIS M.Sc TMsis, Department of Geography, Unversity of Edinburgh, 1991.
[10] B. Lowekamp, N. Miller, D. Sutherland, T Gross, P. Steenkiste and J. Subhlok, A resource monitoring
188
system for network-aware applications, Proc. of 7ih IEEE International Symposium on High Performace Distributed Computing {HPDC), IEEE, July(1998), 189-196.
[ 11) B. Lowekamp D. O' hallaron and T.Gross, Direct queries for discovering network resource properties in a distributed environment, Proc. of the f3'h IEEE International Symposium on High performance Distributed Computing(HPDC), (August 1999). 38-46.
[12) D. Papadias, N. Mamoulis and V.: Delis, Algorithms for querying by spatial structure, VLDB, New York City, New York, USA, (August 1998), 546-557.
[ 13) Z.R. Peng and M.H. Tsou, Internet GIB-Distributed geographic infornu1tion services for the internet and wireless networks, John \lliley & Sons, New Jersey. Post GIS {2005). Post GIS web site. URL http:/postgis. refractions. net.I L&"t checked on 29th September 2004 .
[14] A.J. Soper, C. Walshaw and M. Cross, A combined Evolutionary Search and Multilevel Optimisation Approach to Graph Partitioning, Tech Rep.00/IM/58, Comp. Math.Sci. Univ. Greenwich, London, SElO 9LS, UK, (April 2000).
Jnanabha, Vol. 37, 2007 (Dedicated to Honour Professor S.P Singh on his 7Qth Birthday)
LIE THEORY AND BASIC CLASSICAL POLYNOMIAL By
Kishan Sharma NRI Institute of Techonology and Management,
Gwalior-474001, Madhya Pradesh, India E-mail :[email protected]
and Renu Jain
School of Mathematics and Allied Sciences, Jiwaji University, Gwalior-474009, Madhya Pradesh, India
E-mail : [email protected]
(Received : October 5, 2007)
ABSTRACT
189
In the present paper, an attempt has been made to bring basic hype:rgeometric functions with in the perview of Lie theory by constructing a
dynamical symmetry algebra of basic hypergeometric function 2 <I> 1 • Multiplier
representation theory is then used to obtain generating function for basic analogues of Gegenbauer polynomials. 2000 Mathematics Subject Classification : Primary 20G05, 60E07;
Secondary 14Ml 7, 33C55. Keywords : Lie theory, Gegenbauer polynomials
1. Introduction. The q-analogue of the Gauss function or Heine's series [1.p.259, eqn{U] may be written as
2 <1>1(a,b;c;q;x)= I[a;q,n][b;q,n]xn 
Here [a;q,n] and [n;q]! are respectively the basic Pocchammer's symbol and basic
factorial function defined as [a;q,n]= [a;q][a+l;qJ .. [a+n-l;q] and
[n;q]l= [1;q][2;q] ... [n;q].
The basic differential operator Bg,x is defined [1, p.259, eqn (2)] by the relation
BA q,:r<I>(x) = {<I>(qx)-<I>(x )}/ x(q-1) ... (1.1)
2. The Dynamical Symmetry Algebra of 2 <1> 1 • The dynamical symmetry
190
algebra of the hypergeornetric function has been defined by 1vfiller[2]. We use the
same technique to define the dynamical symmetry algebra of 2 <!> 1 •
Let <D a.ib.q = {rq (y - a)rq (a)/rq(y )}x 2<.D1 (a,j3; y;q;x ~aull{r ... (2.1)
be the basis elements of a subspace of analytical functions of four variables x, s, u
and t, associated with Heine's basic hypergeometric fucntions of Heine's series,
2 <D 1 • Introduction of variables s, u and t renders differential operators independent
of parameters a.,j3 and y and thus facilitates their repeated operation.
The dynamical symmetry algebra of 2 <D 1 is a 15-dimensional complex Lie
algebra isomorphic to sl(4), generated by twelve E"-operators termed as raising
or lowerillg the corresponding suffix in <P aJl,r.<L.
The E"-operators is EA-a,q =s-1 (x(l-x)B~q,x +TB~q.t -sB~q.s -xuB~q,a) ... (2.2)
The action of this operator on ct>a.!l;r,q is given by
A -[ -1• ]<Pl E -a,q<Pa,13,y,q - a ,q a,q,13,y,q ... (2.3)
Twelve E"-operators together with three maintenance operators Jai~Jlff,.J:: and
Identity operator I form a basis for gl(4) ~ sl(4~f}, where
Lie algebra generated by I.
Here J"' ... q=sBAq,x;J,.,..f'>,q=uBAq,u.;JAy,q =tBAq.t andl"=I
with the results
J,,.... a,q <Da,l),y,q = [a;qJPa,!);1,q;J,,.... fl,q cflaJl,y,q = [f3;qfP,:::.j3,;,q
J"' ct> = [y- q "Irr. and I,.,_ ct> =ct> y,q a,fl,y,q ' !¥ a,fl,y,q a,j3,y,q aJl;1,q
is the I-dimensional
... (2.4)
(2.5)
3. The Generating Function for Basic Analogues of Gegenbauer
Polynomial. The action of one parameter subgroup ( expq aE"·-a,q) generated by
the operator E"' -a,q defined in (2.2) on <Da,f3,:•,q defined in (2.1) can be computed by
the multiplier representation method.
L~ing the technique of Miller [3] it can be seen that the transformations
x ~ xs/(a(x-l)+s),s ~ s-a,u ~ u(s-a)!(a(x-l)),t ~ st/(s-a) ... (3.1}
determine the action. Hence the action of one parameter subgroup is given by
191
(exp0
aE"" , -a.4 (y - o:)rq (o:)/rq (y )}x 2 ¢ 1[a,13; y;q;xs J(a(x -1)+ s )]
x(s-a}" -a)/(a(x- (st (s- ... (3.2)
On the other band by expansion, we get
(expqaE"-e<.q·1P,L!l.r.q = f,am[o:-m;qk h-a + m)rq{o:-m)!rq(y) m=G
x <D ro:.q-m f.l •• ,.Q. :::Jscr-rnuf>(! 2 i r , p, : ., .Ji ~ ~ (3.3)
Equating these two values of (expq aE"'-a,q}Pa,!l;~.q we get identity
(s -a )°'+fl-I (a(x -1)-'- s tf· s·r-n x 2¢ 1 [a, (3;y;q;xs /(a(x -1)+ s )]
x
= )' J....J
[·r-a;qJm ![m;q]!}x 2<t>1[a.q-m ,B;y;q;x]s-m ... (3.4) nt=O
For example, Putting ct__,. O,s _:;. l,[3 _,.I.+ m, y-+ 1/2,q .._:;. q 2 and then x _,. x 2 , we
get
(l-a}'--12(1-a + ax2 t° = I ~m[l/2;qJm(l/2;q]/[m;q]!}x 2<t>1[-m,'A+m;l/2;q2 ;x2 }.
m=O
By the definition of basic Gegenbauer polynomials [ 4]
Cim(q;x)= 2 <1> 1 m).+m;l 2;q2 ;x2],
Using {3.6) in (3.5), we get the generating function
(I-af--li2 (1-a +ax2 =} ~
[1 2;qL(l/2;q]/[m;q]!}xC~ni(q;x) .rn::::;c{J
for basic Gegenbauer polynomials.
REFERENCES [ll Kishan Sharma and Renu Jain: Proc. Nat. Acad. Sci. India 77(A), III, (2007) [2] WJr. Miller, Lie Theory and Special Functions, Academic Press. New York, 1968.
[3] W Jr. ~filler, SIAM. J Appl. Math., 25 No. 2, (1973),
{3.5)
(3.6)
(3.7)
[4] H. Exton, q-Hypergeometric functions and Applications, Ellis Horwood
Limited, John Wiley and Sons, New York, Brisbane, Chichester and Toronto,
1983.
Statement of ownership and particulars about the journal
JNAt"JABHA 1. Place of Publication
2. Periodicity of Publication
3. Printer's Name Nationality Address
4. Publisher's Name
Nationality Address
5. Editor's Name Nationality
Address
6. Name and address of the individuals who own the journal and partners of share holders hnlding more than one percent of the total capital
} Address:
}
D. V. Postgraduate College Orai-285001, U.P, India
Annual
1\fr. Dheeraj Gupta Indian Customer Gallery, Orai Orai-285001, U.P, India Mobile : 9450296634
Dr. R.C. Singh Chandel For Vijnana Parishad of India
Indian D.V Postgraduate College Orai-285001, U.P, India
Dr. R.C. Singh Indian
D.V: Post.graduate College Orai- 285001, U.P, India
Vijnana Parishad of India
D.V Postgraduate College Orai-285001, U.P, India
I, Dr. R. C. Singh Chandel hereby declare that the particulars given above are true to the best of my knowledge and belief.
~ Dr. R.C. Singh Ch~ndel
Publisher/Editor
INFORMATION FOR CONTRIBUTORS/SUBSCRIBERS Jn fu"l abha is published annually. It is an interdisciplinary journal devoted primarily to
research articles in all areas of the mathematical, physical and biological sciences; it nn·DnO>r encourage original mathematical works which are motivated by and relevant
aµµu1,;Ct;,nm:> in the social, management, biomedical or engineering sciences. Papers intended blicaticm in this journal should be in typed form or offset-reproduced (not dittoed), A4
size double spaced with generous margins and they may be written in Hindi or English. Manuscripts in duplicate (hard copy along -w-ith soft copy typed in MS wordWage Maker), mentioning 2000 Mathematics Subject Classification, Key Words and authors E-Mail addresses also on front page, if possible, may be submitted to either of the Editors. It is mandatory tor ei:ery author in Jn an ab ha to be a member of the VijnanaParishad
and to pay pu.blication charges which will be based upon the number !pages~
The submission of a paper implies the ati!J::g;r-'_s assurance that the paper has not been widely circulated, copyrighted, published or submitted for publication elsewhere.
Authors are advised to submit only neatly (and carefully) type-wrftten and th:oughly checked manuscripts employing accepted conventions ofreferences, notations, dispiays, etc.; all typescripts not in a format suitable for publication in this journal will be returned unrefereed.
25 .reprints are provided free for each article; additional reprints, in multiples of 25 may be obtained at cost if the authors give prior intimation while returning proofs.
Effecti\-e ;'iith Volume 34 the price per volume is Rs. 200.00 (or US. $25.00). Individual Parishad of India are entitled to free subscriptions to the current
issues oftf-ijs Individual membership: Rs. 100.00 (or US. $20.00) per calender year; Life memibE:rship: Rs. 1000.00 or CS $200.00). Back volumes are available at special price. iPay7n.enr~~cheques must include appropriate bank charges_)
[By a recip:rocity~,;,_greemen~ •vjth the American Mathematical Society, an individual/ life member of the Parishad residing outside North American continent may join the Society by submitting an application for membership on the form that may be obtained from the office of the Society (P.O. Box 6248, Providence, Rhode, Island 02940, U.S.A.) and by pa:.ying the Society's current dues as a reciprocity member at a considerably reduced rate; the usual requirements that the applica1lt be endorsed by tvw members of the Society and that the candidate be elected the Council are waived, but this reduction in dues to the Society does not apply to those individualiife members of the Parishad who reside, even on a temporary basis, in the North American area \i.e., ES.A and Canada).
The mathematical content of tl1is journal is indexed in the Society's biweekly Current Mathematical Publications a.'1d is reviewed, among others, by Mathematical Reviews !US.A), Zentralblatt fiir Mathematik (Germany) and Applied Mechanics Reviews (US.A).]
All cow..munications regarding subsriptions, order for back volumes, membership of the Vyniina Pa:rishad of India, change of address, etc. and all books for review, should be addressed to_
The Secretary Vij iiana Parishad of India
D.V Postg:raduat e College Orai 285001, UP., India or DA-77, Sheesh Mahal Apartments, Shalimar Bagh, Delhi-88
E-Mail :re_ [email protected] Phone:09811488921
CONTENTS PROFESSOR S.P. SINGH: AS A MA.i."'l Ai"'ID MATIIEMATICL\...~
-R.C. Singh Chandel
~#·
A CORRECTION TO AN EXAMPI_,E OF RE!\"U CHUG M'D SANJAY KUMAR FOR WEAKLY COMPATIBLE SELF-MAPS
-T. Phaneedra A NOTE ON PAIRWISE SLIGHTLY SEMI-CONTINUOUS FUNCTIONS
- M.C.Sharma and V.K.Solanki
N-POLJCY FOR M!G!I MACHINI<:; REPAJR PROBLEM WITH STA.i.-...rDBY AND COMMON CAUSE FAILURE -M. Jain and Satya Prakash Gautam
ON GENERATING RELATIONSHIPS FOR FOX'S H-FUNCTION AND MULTIVARIABLE H-FUNCTION -8.8. Jaimini and Hemlata Saxena
ON l)J<~COMPOSITIONS OF PROJECTIVE CURVATURE TENSOR IN CONFORMAIJ FINSLER SPACE -C.K. Mishra and Gautam Lodhi EXTENSION OF P-TRNSFORM TO A CLASS OF GE:r-.t'J<::RALISED FUNCTION
-M.M.P. Singh and Neeraj kumar ON APPROXIMATION OF A FUNCTION BY GENER.i\.LIZED NORl,lJND MEANS O:F ITS .FOURIER SERIES
-Ashuthosh Pahak, Murtaza Turabi and Tushar Kant Jhala A COMMON FIXED POINT THEOREM IN MENGER SPACES
-R.C.Dimri and N.S.Gariy2
PERFORMANCE PREDICTION OF MIXED MULTICOMPONENTS MACHINING SYSTEM WITH BALKING, RENEGING, ADDITIONAL REPAIRMEN -M.Jain and G.C. Sharma and Supriya Maheshwari
A NOTE ON MHD RAYLEIGH FLOW OF A f'LUID OF EQUAL KINEMATIC VISCOSITY Al'li'D MAGNETIC VISCOSITY PAST A PERFECTLY CONDUCTING
... 3-20
... 21-22
.. .23-28
... 29-38
... 39-18
.. .19-62
... 63-68
... 69-75
... Ti-82
.. B:J-%
POROUS PLATE -P.K. Mittal, Mukesh Bijafwan and Vijay Dhasmana ... 99-JOO
ON SOME NEW MULTIPLE HYPERGE01"IETRIC FUNCTIONS RELATED TO LAURICELIA'S FUNCTIONS -R.C. Singh Chandel and Vandana Gupta ... J 07- l 22 CERTAINMULTIPI.,EINTEGHALTRAi.'l'SFOlli'\IATIONSPERTA-'-..,.l:'IGTOTHE MULTIVAllIABLE A-FUNCTION
-V.8.L.Chaurasia and Pinky Lata ... 123-134
ON UNIFROilM T (C,l) SUMMABILITY OF LEGENDRE SEH.IES -V.N. Tripathi and Sonia Pal ... 135-111
FINITE M/G/1 QUEUEING SYSTEM WIT.II BERNOULLI FEEDBACK UNrnm OP'ITMAl_, N-POLICY -M.R. Dhakad, Madhu Jain and G.C. Sharma ... 143-15"1 g-REGULAllLY ORDERED SPACES
-M.C. Sharma and N. Kumar ... 155-159 ON TIIE DEGREE OF APPROXIlVIATION OF ETJLER'S 11,:lF.,ANS
-Ashutosh Pathak and Namrata Choudhary ... 161-166 MULTIVARIABLE ANALOGUES OF GENERALIZED TRUESDEl.L POLYNOMIAl_,S ·R.C. Singh Chandel and K.P. Tiwari ... 167-176 SP-QUADTREE : AN APPROACH OF DATA STRUCTURING FOR PAllALLELIZATION OF SPATIAL DATA
-N.Badal, Krishna kant and G.K. Sharma ... l 77- l 8fi LIE THEORY ANl..) BASIC CLASSICAL POLYNOMIAL
-Kishan Sharma and Renu Jain ... 189-191
