Volume 102, Issue 2
August 1992, pages 83-158
pp 83-91 August 1992
Proof of some conjectures on the mean-value of Titchmarsh series — III
R Balasubramanian K Ramachandra
With some applications in view, the following problem is solved in some special case which is not too special. LetF(s) =Σ_{n}^{∞}=1a_{n}λ_{n}^{−s} be a generalized Dirichlet series with 1 =λ_{1} <λ_{2} < …,λ_{n}≤Dn, andλ_{n+1} -λ_{n} ≥D^{− 1}λ_{n+1}^{− a} where α>0 andD(≥ 1) are constants. Then subject to analytic continuation and some growth conditions, a lower bound is obtained for$$(1/H)\int {_O^H |} F(it)|^2 dt$$. These results will be applied in other papers to appear later.
pp 93-123 August 1992
Combinatorial meaning of the coefficients of a Hilbert polynomial
In [1] Abhyankar defines an idealI(p, a) generated by certain minors of a matrixX, the entries ofX being independent indeterminates, and proves that the Hubert function ofI(p, a) coincides with its Hilbert polynomialF(V) and obtains it in the form$$F(V) = \sum\limits_{D \geqslant 0} {( - 1)} ^D F_D (m,p,a)\left( \begin{gathered} c - D + V \hfill \\ V \hfill \\ \end{gathered} \right)$$. He also proves thatF(V) is the number of certain “indexed” monomials of degreeV in the entries ofX and that the coefficientsF_{D}(m,p,a) are non-negative integers and asks for their combinatorial meaning. In this paper we characterize the indexed monomials in terms of certain sets of lattice paths, called frames, and prove that the coefficientsF_{D}(m, p, a) count certain families of such frames.
pp 125-128 August 1992
A note on two absolute summability methods
In this paper we have established a relation between ¦R,p_{n};δ¦_{k} and ¦R,q_{n};δ¦_{k} summability methods which generalizes the results of Bor [1] and Bosanquet [2].
pp 129-139 August 1992
On deficiencies of differential polynomials
Anand Prakash Singh Raj Shree Dhar
Bounds for deficiencies of differential polynomials have been given and some relations between Nevanlinna characteristic of differential polynomials inf and Nevanlinna characteristic function off have been obtained.
pp 141-146 August 1992
A note on the identity operators of fractional calculus
The application of an identity operator for Saigo’s fractional calculus operators is shown by evaluating the limit of an indeterminate form. Its special case yields the result which has been used as an infinitesimal generator in the semigroup theory. Also, an identity operator for the recently introduced multi-dimensional fractional operators (due to Srivastava and Raina [8]) is discussed.
pp 147-153 August 1992
An RSA based public-key cryptosystem for secure communication
A new cryptosystem that uses modulo arithmetic operations is proposed. It is based on Rivest-Shamir-Adleman’s public key cryptosystem. A feature of the proposed system is that the encryption and decryption procedures are computationally less intensive, and hence the system is amenable for high data bit rate communications.
pp 155-158 August 1992
A note to the paper “An efficient algorithm for linear programming” of V Ch Venkaiah
In Venkaiah [1] an algorithm for solving linear optimization problems based on the idea of the projective algorithm of Karmarkar, is proposed. The essential simplification in the new algorithm is the use of a fixed projection operator. In this way the algorithm requires onlyO(n^{2}) operations to obtain a sufficient exact solution. In this note it is shown that in some special cases the algorithm of Venkaiah yields a feasible solution that is far from the optimal one.
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