iosr-Jm   Volume-7 ~ Issue-5

Abstract: In this paper we established Boundedness and Compositions of the Operator 𝐺𝜌 ,𝜂 ,𝛾 ,𝜔 ;𝑎+𝛹 (𝑥) and the Inversion Formulae on the space L(a,b) and C[a,b] , given by Hartely and Lorenzo[5]

Key words -𝐺𝜌 ,𝜂 ,𝑟 [𝑎, 𝑧] function, Riemann liouville fractional integral, Riemann liouville fractional derivative , beta-integral.

[1]. Carl F. Lorenzo, Tom T. Hartley: Generalized Functions for the Fractional Calculus, NASA/TP—1999-209424/REV1(1999)
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[5]. Lorenzo, C.F. and Hartely, T.T. : R-function relationships for application in the fractional calculus, NASA Tech. Mem. 210361, 1-22,
(2000)

Abstract: Neighbour Balanced Block Designs, permitting the estimation of direct and neighbour effects, are used when the treatment applied to one experimental plot may affect the response on neighbouring plots as well as the response on the plot to which it is applied. The allocation of treatments in these designs is such that every treatment occurs equally often with every other treatment as neighbours. Neighbour Balanced Block Designs for observations correlated within a block have been investigated for the estimation of direct as well as left and right neighbour effects of treatments. It is observed that efficiency for direct as well as neighbour effects is high, in case of Complete block designs i.e., m  0 for Nearest Neighbour correlation structure with  in the interval 0.1 to 0.4. In case of incomplete block designs m 1,2,, v  4 for Nearest Neighbour correlation structure turns out to be more efficient as compared to others models with  in the interval 0.1 to 0.4. The performance of Nearest neighbour balanced block designs is satisfactory for ARMA(1,1) models. The gain in efficiency of NNBD and NNBIBD over regular block design is high under MA(1) models for direct and neighbour effects of treatments.

Keywords: Neighbour Balanced Block Design; Correlated observations; Generalized least squares; AutoRegressive; Moving Average; Nearest neighbour; Efficiency; Regular Block Design.

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Abstract: In this paper we have established a theorem on k   |C,, , , | -summability factor, which gives some new results.

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