N. SUBRAMANIAN 1, U. K. MISRA 2

Transcription

1 TWMS J App Eng Math V, N, 0, pp THE INVARIANT χ SEQUENCE SPACES N SUBRAMANIAN, U K MISRA Abstract In this paper we define inariatness of a double sequence space of χ and examine the inariatness of the double sequence space of χ Furthermore, we gie duals of double sequence space of χ Keywords: Gai sequence, analytic sequence, modulus function, double sequences AMS Subject Classification: 40A05,40C05,40D05 Introduction Throughout w, χ and Λ denote the classes of all, gai and analytic scalar alued single sequences, respectiely We write w for the set of all complex sequences x mn ), where m, n N, the set of positie integers Then, w is a linear space under the coordinate wise addition and scalar multiplication Some initial work on double sequence spaces is found in Bromwich [4] Later on, they were inestigated by Hardy [5], Moricz [9], Moricz and Rhoades [0], Basarir and Solankan [], Tripathy [7], Turkmenoglu [9], and many others Let us define the following sets of double sequences: M u t) := x mn ) w : sup m,n N x mn t mn < }, C p t) := x mn ) w : p lim m,n x mn l t mn = for some l C }, C 0p t) := x mn ) w : p lim m,n x mn t mn = }, L u t) := x mn ) w : m= n= x mn t mn < }, C bp t) := C p t) M u t) and C 0bp t) = C 0p t) M u t); where t = t mn ) is the sequence of strictly positie reals t mn for all m, n N and p lim m,n denotes the limit in the Pringsheim s sense In the case t mn = for all m, n N; M u t), C p t), C 0p t), L u t), C bp t) and C 0bp t) reduce to the sets M u, C p, C 0p, L u, C bp and C 0bp, respectiely Now, we may summarize the knowledge gien in some document related to the double sequence spaces Gökhan and Colak [,] hae proed that M u t) and C p t), C bp t) are complete paranormed spaces of double sequences and gae the α, β, γ duals of the spaces M u t) and C bp t) Quite recently, in her PhD thesis, Zelter [3] has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences Mursaleen and Edely [4] hae recently introduced the statistical conergence and Cauchy for double sequences and gien the relation Department of Mathematics, SASTRA Uniersity, Thanjaur-63 40, India, nsmaths@yahoocom Department of Mathematics and Statistics, Berhampur Uniersity, Berhampur ,Odissa, India, umakanta misra@yahoocom Manuscript receied 0 August 0 TWMS Journal of Applied and Engineering Mathematics Vol No c Işık Uniersity, Department of Mathematics 0; all rights resered 73

2 74 TWMS J APP ENG MATH V, N, 0 between statistical conergent and strongly Cesàro summable double sequences Nextly, Mursaleen [5] and Mursaleen and Edely [6] hae defined the almost strong regularity of matrices for double sequences and applied these matrices to establish a core theorem and introduced the M core for double sequences and determined those four dimensional matrices transforming eery bounded double sequences x = x jk ) into one whose core is a subset of the M core of x More recently, Altay and Basar [7] hae defined the spaces BS, BS t), CS p, CS bp, CS r and BV of double sequences consisting of all double series whose sequence of partial sums are in the spaces M u, M u t), C p, C bp, C r and L u, respectiely, and also hae examined some properties of those sequence spaces and determined the α duals of the spaces BS, BV, CS bp and the β ϑ) duals of the spaces CS bp and CS r of double series Quite recently Basar and Seer [8] hae introduced the Banach space L q of double sequences corresponding to the well-known space l q of single sequences and hae examined some properties of the space L q Quite recently Subramanian and Misra [9] hae studied the space χ M p, q, u) of double sequences and hae gien some inclusion relations Spaces are strongly summable sequences was discussed by Kuttner [3], Maddox [3], and others The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox [8] as an extension of the definition of strongly Cesàro summable sequences Connor [33] further extended this definition to a definition of strong A summability with respect to a modulus where A = a n,k ) is a nonnegatie regular matrix and established some connections between strong A summability, strong A summability with respect to a modulus, and A statistical conergence In [34] the notion of conergence of double sequences was presented by A Pringsheim Also, in [35]- [38], and [39] the four dimensional matrix transformation Ax) k,l = m= n= amn kl x mn was studied extensiely by Robison and Hamilton In their work and throughout this paper, the four dimensional matrices and double sequences hae real-alued entries unless specified otherwise In this paper we extend a few results known in the literature for ordinarysingle) sequence spaces to multiply sequence spaces We need the following inequality in the sequel of the paper For a, b, 0 and 0 < p <, we hae a + b) p a p + b p ) The double series m,n= x mn is called conergent if and only if the double sequence s mn ) is conergent, where s mn = m,n i,j= x ijm, n N) see[]) A sequence x = x mn )is said to be double analytic if sup mn x mn /m+n < The ector space of all double analytic sequences will be denoted by Λ A sequence x = x mn ) is called double gai sequence if m + n)! x mn ) /m+n 0 as m, n The double gai sequences will be denoted by χ Let φ = all finite sequences} Consider a double sequence x = x ij ) The m, n) th section x [m,n] of the sequence is defined by x [m,n] = m,n i,j=0 x iji ij for all m, n N ; where I ij denotes the double sequence whose only non zero term is a i+j)! in the i, j) th place for each i, j N An FK-spaceor a metric space)x is said to hae AK property if I mn ) is a Schauder basis for X Or equialently x [m,n] x An FDK-space is a double sequence space endowed with a complete metrizable; locally conex topology under which the coordinate mappings x = x k ) x mn )m, n N)

3 N SUBRAMANIAN, U K MISRA: THE V INVARIANT χ SEQUENCE SPACES 75 are also continuous If X is a sequence space, we gie the following definitions: i)x = the continuous dual of X; ii)x α = a = a mn ) : m,n= a mn x mn <, for each x X } ; iii)x β = a = a mn ) : m,n= a mn x mn is conegent, for each x X } ; i)x γ = a = a mn ) : sup mn } M,N m,n= a mnx mn <, for each x X ; )let X beanf K space φ; then X f = fi mn ) : f X } ; } i)x δ = a = a mn ) : sup mn a mn x mn /m+n <, for each x X ; X α X β, X γ are called α orköthe T oeplitz)dual of X, β or generalized Köthe T oeplitz) dual ofx, γ dual of X, δ dual ofx respectielyx α is defined by Gupta and Kamptan [0] It is clear that x α X β and X α X γ, but X β X γ does not hold, since the sequence of partial sums of a double conergent series need not to be bounded The notion of difference sequence spaces for single sequences) was introduced by Kizmaz [30] as follows Z ) = x = x k ) w : x k ) Z} for Z = c, c 0 and l, where x k = x k x k+ for all k N Here c, c 0 and l denote the classes of conergent,null and bounded sclar alued single sequences respectiely The difference space b p of the classical space l p is introduced and studied in the case p by BaŞar and Altay in [4] and in the case 0 < p < by Altay and BaŞar in [43] The spaces c ), c 0 ), l ) and b p are Banach spaces normed by x = x + sup k x k and x bp = k= x k p ) /p, p < ) Later on the notion was further inestigated by many others We now introduce the following difference double sequence spaces defined by Z ) = x = x mn ) w : x mn ) Z } where Z = Λ, χ and x mn = x mn x mn+ ) x m+n x m+n+ ) = x mn x mn+ x m+n + x m+n+ for all m, n N Definitions and Preliminaries Let = mn ) be any fixed sequence of nonzero } complex numbers satisfying Λ = x = x mn ) : sup mn mn x mn /m+n < } χ = x = x mn ) : m + n)! mn x mn ) /m+n 0 as m, n In this paper Λ and χ will denote the sequence spaces of Pringsheim sense double analytic inariant and Pringsheim sense double gai inariant sequences respectiely The space Λ is a inariant metric space with the metric } dx, y) = sup mn mn x mn mn y mn /m+n : m, n :,, 3, )

4 76 TWMS J APP ENG MATH V, N, 0 forall x = x mn } and y = y mn }in Λ The space χ is a inariant metric space with the metric } dx, y) = sup mn m + n)! mn x mn mn y mn ) /m+n : m, n :,, 3, 3) forall x = x mn } and y = y mn }in χ Definition A sequence X is inariant if X = X where X = x = x mn ) : mn x mn ) X}, where X = Λ and χ In this paper we define inariantness of a sequence space X and gie necessary and sufficient conditions for Λ and χ to inariant Now, if X = Λ or χ is inariant sequence spaces then we hae the following results 3 Main Results Theorem 3 Let χ be a inariant sequence space Then i) χ is a Banach inariant space if and only if χ is a Banach inariant metric space, ii) χ is separable if and only if χ is separable Proof Let u = u mn ) and = mn ) be any fixed sequence of nonzero complex numbers such that lim mn sup m + n)! u mn 0 ) /m+n and lim mn sup m + n)! mn 0 ) /m+n are positie may be infinite) If mn = λ for eery m, n, then obiously χ is inariant, where λ is a scalar This completes the proof Theorem 3 Let w mn = u mn mn for each m, n N, where mn = mn Then i) χ χ u if and only if sup mn w mn < ii) χ = χ u if and only if 0 < inf mn w mn w mn sup mn s mn < Proof Sufficiency is triial, since u mn x mn /m+n = w mn /m+n mn x mn /m+n 4) For the necessity suppose that χ χ u but sup mn = Then there exists a strictly increasing sequence w mi n i ) > i we put 0 if m, n m + n)! x mn mn ) /m+n mi n i = i 5) u mi if m, n = m n i n i i Then we hae m + n)! x mn mn ) /m+n < and m + n)! x mn u mn ) /m+n = i, where m, n = m i n i When x χ χ u contrary to the assumption that χ χ u ii) To proe this, it is enough to show that χ u χ if and only if inf mn w mn > 0 It is obious that inf mn w mn > 0 if and only if sup mn w mn < Hence the result follows from proof i) Theorem 33 i) χ χ if and only if sup mn mn <, ii) χ χ if and only if inf mn mn > 0, iii)χ = χ if and only if 0 < inf mn mn mn sup mn mn <

5 N SUBRAMANIAN, U K MISRA: THE V INVARIANT χ SEQUENCE SPACES 77 Proof Taking = Theorem 3 i) upto m, n)th term and replacing u by in It is triial that inf mn mn > 0 if and only if sup mn mn < Hence taking u = upto m, n)th term in Theorem 3 i), we get Theorem 33ii) Finally, taking Taking u = upto m, n)th term in Theorem 3 ii), since clearly inf mn mn > 0 if and only if sup mn mn <, we get iii) Corollary 3 If χ is inariant if and only if 0 < inf mn mn mn sup mn mn < Proof Follows from Theorem 33 iii) Theorem 34 i) χ χ u if and only if w = w mn ) χ, ii) χ = χ u if and only if w / χ Proof i) The sufficiency is triial by an equation 4) For the necessity suppose that χ χ u but w / χ ) Then, either w Λ or)w / Λ Now we put m + n)! x mn ) /m+n = w mn = Then u mn) /m+n mn) /m+n m + n)! x mn mn ) /m+n = and m + n)! x mn u mn ) /m+n = w mn ) Whence x χ χ u, contrary to the assumption that χ χ u Hence we obtain the necessity ii) Sufficiency, let w χ χ u by i) Let x χ u, so that m + n)! x mn u mn ) /m+n χ Now, since w χ, lim mn w mn = 0 Therefore, from the equality ) m + n)! x mn mn ) /m+n /m+n = m + n)! x mn u mn w mn, we hae m + n)! x mn mn ) /m+n χ and hence χ u χ Necessity: Suppose that χ = χ u, that is χ χ u and χ u χ Then lim mn w mn = lim mn u mn mn and lim mn w mn = lim mn u mn mn = 0 It is triial that lim mn w mn = 0 if and only if lim mn w mn 0 Hence w / χ Theorem 35 i) χ χ if and only if χ ii) χ = χ if and only if / χ and lim mn mn 0

6 78 TWMS J APP ENG MATH V, N, 0 Proof Taking = and replacing u by in Theorem 34 i), we obtain i) Theorem 34 ii) gies us ii) for u = Remark 3 If χ and lim mn mn = 0 that is χ, then χ χ Proposition 3 χ Γ Proof Let x χ Then we hae m + n)! x mn mn ) /m+n 0 as m, n Here, we get x mn mn /m+n 0 as m, n Thus we hae x Γ and so χ Γ Proposition 3 Γ ) β Λ Proof Let y = y mn ) be an arbitrary point in Γ ) β If y is not in Λ, then for each natural number p, we can find an index m p n p such that ymp n p /m p +n p > p mn, p =,, 3, ) 6) Define x = x mn } by x mn = p m+n mn, 0, otherwise Then x is in Γ, but for infinitely mn, for m, n) = m p, n p ) for somep N 7) y mn x mn > 8) Consider the sequence z = z mn }, where z = x s with s = x mn mn, z mn = x mn mn 9) m= n= Then z is a point of Γ Also, z mn = 0 Hence, z is in Γ ; but, by 8), z mn y mn does not conerge: x mn y mn dierges 0) Thus, the sequence y would not be in Γ ) β This contradiction proes that Γ ) β Λ ) Let y n n = x n n = and y mn mn = x mn mn = 0 m > ) for all n, then obiously x Γ and y Λ, but x mn y mn = Hence, y / Γ β ) ) m= n= From ) and ), we are granted Γ ) β Λ Proposition 33 The β dual space of χ is Λ

7 N SUBRAMANIAN, U K MISRA: THE V INVARIANT χ SEQUENCE SPACES 79 Proof First, we obsere that χ Γ, by Proposition 3 Therefore Γ ) β χ ) β But Γ ) β Λ, by Proposition 3 Hence Λ χ ) β 3) Next we show that χ ) β Λ Let y = y mn ) χ ) β Consider f x) = m= n= x mny mn with x = x mn ) χ x = [I mn I mn+ ) I m+n I m+n+ )] 0, 0, 0, 0, 0 0, 0, 0, 0, 0 = 0, 0,,, 0 m+n)! mn) /m+n m+n)! mn) /m+n 0, 0, 0, 0, 0 0, 0, 0, 0, 0 0, 0, 0, 0, 0 0, 0,,, 0 m+n)! mn) /m+n m+n)! mn) /m+n 0, 0, 0, 0, 0 m + n)! x mn mn ) /m+n} = Hence conerges to zero Therefore [I mn I mn+ ) I m+n I m+n+ )] χ Hence d I mn I mn+ ) I m+n I m+n+ ), 0) = But 0, 0, 0, 0, 0 0, 0, 0, 0, 0 0, 0,,, 0 m+n)! mn) /m+n m+n)! mn) /m+n 0, 0,,, 0 m+n)! mn ) /m+n m+n)! mn ) /m+n 0, 0, 0, 0, 0 y mn mn /m+n f d I mn I mn+ ) I m+n I m+n+ ), 0) f < for each m, n Thus y mn ) is a double inariant bounded sequence and hence an inariant analytic sequence In other words y Λ But y = y mn ) is arbitrary in χ ) β Therefore χ ) β Λ 4) From 3) and 4) we get χ ) β = Λ

8 80 TWMS J APP ENG MATH V, N, 0 Proposition 34 Λ dual of χ is Λ Proof Let y Λ dual of χ Then x mn y mn M m+n mn for some constant M > 0 and for each x χ Therefore y mn mn M m+n for each m, n by taking 0, 0, 0, 0, 0 0, 0, 0, 0, 0 x = I mn = 0, 0,, 0, 0 m+n)! mn) /m+n 0, 0, 0, 0, 0 This shows that y Λ Then χ ) Λ Λ 5) On the other hand, let y Λ Let ɛ > 0 be gien Then y mn mn < M m+n for each m, n and for some constant M > 0 But x χ Hence m + n)! x mn mn ) < ɛ m+n for each m, n and for each ɛ > 0 ie x mn < ɛ m+n m+n)! mn) /m+n Hence y χ ) Λ x mn y mn = x mn y mn < ɛ m+n m+n)! mn) /m+n M m+n = ɛm) m+n m+n)! mn) /m+n Λ χ ) Λ 6) From 5) and 6) we get χ ) Λ = Λ Proposition 35 Let χ ) denote the dual space of χ Then we hae χ ) = Λ Proof We recall that 0, 0, 0, 0, 0 0, 0, 0, 0, 0 x = I mn = 0, 0,, 0, 0 m+n)! mn ) /m+n 0, 0, 0, 0, 0 with in the m, n) th position and zero otherwise, with m+n)! mn ) /m+n x = I mn, m + n)! x mn mn ) /m+n}

9 N SUBRAMANIAN, U K MISRA: THE V INVARIANT χ SEQUENCE SPACES 8 0 /, 0, 0, 0, 0 /+n = ) /m+n 0 /m+, 0, m+n)!mn m+n)! mn, 0, 0 /m+n+ 0 /m+, 0, 0, 0, 0 /m+n+ 0, 0, 0, 0, 0 0, 0, 0, 0, 0 = 0, 0, /m+n, 0, 0 0, 0, 0, 0, 0 which is a double χ sequence Hence I mn χ Let us take f x) = m= n= x mny mn with x χ and f χ ) Take x = xmn ) = I mn χ Then y mn mn /m+n f d I mn, 0) < for each m, n Thus y mn ) is a bounded inariant sequence and hence an double analytic inariant sequence In other words y Λ Therefore χ ) = Λ Proposition 36 Λ ) β = Λ Proof Step : Let x mn ) Λ and let y mn ) Λ Then we get y mn mn /m+n M for some constant M > 0 Also x mn mn ) Λ x mn mn ) /m+n ɛ = M Hence m= x mn m+n M m+n mn n= x mny mn m= n= x mn y mn < m= n= m+n M m+n M m+n mn) < m= n= < m+n mn ) Therefore, we get that x mn ) Λ β ) and so we hae Λ Λ ) β 7) Step : Let x mn ) Λ β ) This says that x mn y mn < for each y mn ) Λ 8) m= n= Assume that x mn ) / Λ, then there exists a sequence of positie integers m p + n p ) strictly increasing such that x mp+np > mp+np p =,, 3, ) ) Take and y mp,n p = ) m p+n p p =,, 3, )

10 8 TWMS J APP ENG MATH V, N, 0 Then y mn ) Λ But m= n= x mny mn = p= y mn = 0 otherwise xmpnp y mpnp > We know that the infinite series dierges Hence m= n= x mny mn dierges This contradicts 8) Hence x mn ) Λ Therefore Λ ) β Λ 9) From 7) and 9) we get Λ ) β = Λ References [] Aposto, T, 978), Mathematical Analysis, Addison-Wesley, London [] Basarir, M and Solancan, O, 999), On some double sequence spaces, J Indian Acad Math, ), [3] Bektas, C and Altin, Y, 003), The sequence space l M p, q, s) on seminormed spaces, Indian J Pure Appl Math, 344), [4] Bromwich, TJI A, 965), An introduction to the theory of infinite series, Macmillan and CoLtd, New York [5] Hardy, GH, 97), On the conergence of certain multiple series, Proc Camb Phil Soc, 9, [6] Krasnoselskii, MA and Rutickii, YB, 96), Conex functions and Orlicz spaces, Gorningen, Netherlands [7] Lindenstrauss, J and Tzafriri, L, 97), On Orlicz sequence spaces, Israel J Math, 0, [8] Maddox, IJ, 986), Sequence spaces defined by a modulus, Math Proc Cambridge Philos Soc, 00), 6-66 [9] Moricz, F, 99), Extentions of the spaces c and c 0 from single to double sequences, Acta Math Hung, 57-), 9-36 [0] Moricz, F and Rhoades, BE, 988), Almost conergence of double sequences and strong regularity of summability matrices, Math Proc Camb Phil Soc, 04, [] Mursaleen, M, Khan, MA and Qamaruddin, 999), Difference sequence spaces defined by Orlicz functions, Demonstratio Math, Vol XXXII, [] Nakano, H, 953), Concae modulars, J Math Soc Japan, 5, 9-49 [3] Orlicz, W, 936), Über Raume `L M Bull Int Acad Polon Sci A, [4] Parashar, SD and Choudhary, B, 994), Sequence spaces defined by Orlicz functions, Indian J Pure Appl Math, 54), [5] Chandrasekhara Rao, K and Subramanian, N, 004), The Orlicz space of entire sequences, Int J Math Math Sci, 68, [6] Ruckle, WH, 973), FK spaces in which the sequence of coordinate ectors is bounded, Canad J Math, 5, [7] Tripathy, BC, 003), On statistically conergent double sequences, Tamkang J Math, 343), 3-37 [8] Tripathy, BC, Et, M and Altin, Y, 003), Generalized difference sequence spaces defined by Orlicz function in a locally conex space, J Anal Appl, 3), 75-9 [9] Turkmenoglu, A, 999), Matrix transformation between some classes of double sequences, J Inst Math Comp Sci Math Ser, ), 3-3 [0] Kamthan, PK and Gupta, M, 98), Sequence spaces and series, Lecture notes, Pure and Applied Mathematics, 65 Marcel Dekker, In c, New York [] Gökhan, A and Çolak, R, 004), The double sequence spaces c P p) and c P B p), Appl Math Comput, 57), [] Gökhan, A and Çolak, R, 005), Double sequence spaces l, ibid, 60), [3] Zeltser, M, 00), Inestigation of Double Sequence Spaces by Soft and Hard Analitical Methods, Dissertationes Mathematicae Uniersitatis Tartuensis 5, Tartu Uniersity Press, Uni of Tartu, Faculty of Mathematics and Computer Science, Tartu

11 N SUBRAMANIAN, U K MISRA: THE V INVARIANT χ SEQUENCE SPACES 83 [4] Mursaleen, M and Edely, OHH, 003), Statistical conergence of double sequences, J Math Anal Appl, 88), 3-3 [5] Mursaleen, M, 004), Almost strongly regular matrices and a core theorem for double sequences, J Math Anal Appl, 93), [6] Mursaleen, M and Edely, OHH, 004), Almost conergence and a core theorem for double sequences, J Math Anal Appl, 93), [7] Altay, B and Basar, F, 005), Some new spaces of double sequences, J Math Anal Appl, 309), [8] Basar, F and YSeer, Y, 009), The space L p of double sequences, Math J Okayama Uni, 5, [9] Subramanian, N and Misra, UK, 00), The semi normed space defined by a double gai sequence of modulus function, Fasciculi Math, 46 [30] Kizmaz, H, 98), On certain sequence spaces, Cand Math Bull, 4), [3] Kuttner, B, 946), Note on strong summability, J London Math Soc,, 8- [3] Maddox, IJ, 979), On strong almost conergence, Math Proc Cambridge Philos Soc, 85), [33] Cannor, J, 989), On strong matrix summability with respect to a modulus and statistical conergence, Canad Math Bull, 3), [34] Pringsheim, A, 900), Zurtheorie derzweifach unendlichen zahlenfolgen, Math Ann, 53, 89-3 [35] Hamilton, HJ, 936), Transformations of multiple sequences, Duke Math J,, 9-60 [36] -, 938), A Generalization of multiple sequences transformation, Duke Math J, 4, [37] -, 938), Change of Dimension in sequence transformation, Duke Math J, 4, [38] -, 939), Preseration of partial Limits in Multiple sequence transformations, Duke Math J, 4, [39], Robison, GM, 96), Diergent double sequences and series, Amer Math Soc Trans, 8, [40] Silerman, LL, On the definition of the sum of a diergent series, unpublished thesis, Uniersity of Missouri studies, Mathematics series [4] Toeplitz, O, 9), Über allgenmeine linear mittel bridungen, Prace Matemalyczno Fizyczne warsaw), [4] Basar, F and Altay, B, 003), On the space of sequences of p bounded ariation and related matrix mappings, Ukrainian Math J, 55), [43] Altay, B and Basar, F, 007), The fine spectrum and the matrix domain of the difference operator on the sequence space l p, 0 < p < ), Commun Math Anal, ), - [44] Çolak, R, Et, M and Malkowsky, E, 004), Some Topics of Sequence Spaces, Lecture Notes in Mathematics, Firat Uni Elazig, Turkey, -63, Firat Uni Press, ISBN: N Subramanian, for a photograph and biography, see TWMS Journal of Applied and Engineering Mathematics, Volume, No, 0

12 84 TWMS J APP ENG MATH V, N, 0 U K Misra was born on 0th July 95 He has been working as a faculty in the PG Department of Mathematics, Berhampur Uniersity, Berhampur, Odisha, India since last 8 years 0 scholars hae already been awarded PhD under his guidance and presently 7 scholars are working under him for PhD and DSc degree He has published around 70 papers in arious National and International Journal of repute The field of Prof Misra s research is summability theory, sequence space, Fourier series, inentory control, mathematical modeling He is reiewer of Mathematical Reiew published by American Mathematical Society Prof Misra has conducted seeral national seminars and refresher courses sponsored by UGC India