Volume 107, Issue 1
February 1997, pages 1-100
pp 1-12 February 1997
On partial sums of mock theta functions of order three
The object of this paper is to define and study the properties of partial mock theta functions of order three, on the lines Ramanujan had studied partial θ-functions. These new partial functions have been expressed in terms of basic hypergeometric function_{2}Φ_{1}. Their continued fractions representations have also been given.
pp 13-19 February 1997
On degrees of maps between Grassmannians
Vimala Ramani Parameswaran Sankaran
Let G_{n,k} denote the oriented grassmann manifold of orientedk-planes in ℝ^{n}. It is shown that for any continuous mapf: G_{n,k} → G_{n,k}, dim G_{n,k} = dim G_{m,l} = l(m −l), the Brouwer’s degree is zero, providedl > 1,n ≠ m. Similar results for continuous mapsg: ℂG_{m,l} → ℂG_{n,k},h: ℍG_{m,l} → ℍG_{n,k}, 1 ≤ l < k ≤ n/2, k(n — k) = l(m — l) are also obtained.
pp 21-25 February 1997
Elementary counterexamples to Kodaira vanishing in prime characteristic
Using methods from the modular representation theory of algebraic groups one can construct [1] a projective homogeneous space forSL_{4}, in prime characteristic, which violates Kodaira vanishing. In this note we show how elementary algebraic geometry can be used to simplify and generalize this example.
pp 27-33 February 1997
Mridula Garg Mahesh Kumar Gupta
In this paper we have solved a double convolution integral equation whose kernel involves the product of theH-functions of several variables and a general class of multivariable polynomials. Due to general nature of the kernel, we can obtain from it, solutions of a large number of double and single convolution integral equations involving products of several classical orthogonal polynomials and simpler functions. We have also obtained here solutions of two double convolution integral equations as special cases of our main result. Exact reference of three known results, which are obtainable as particular cases of one of these special cases, have also been included.
pp 35-42 February 1997
On a new geometric property for Banach spaces
In this paper we study a geometric property for Banach spaces called condition (*), introduced by de Reynaet al in [3], A Banach space has this property if for any weakly null sequencex_{n} of unit vectors inX, ifx_{*}^{n} is any sequence of unit vectors inX^{*} that attain their norm at x_{n}’s, then$$x_n^* \mathop \to \limits^{w*} 0.$$. We show that a Banach space satisfies condition (*) for all equivalent norms iff the space has the Schur property. We also study two related geometric conditions, one of which is useful in calculating the essential norm of an operator.
pp 43-55 February 1997
Intrinsic geometry of curves and the Bonnor’s equation
J L López-Bonilla G A Ovando Z J M Rivera-Rebolledo
The exact form of the paths of charged particles moving according to Bonnor’s equation, for the case of constant electromagnetic field, is found. The method employed is fully relativistic and is based on systematic use of the Frenet-Serret equations which determine the world line geometry in Minkowskian space.
pp 57-70 February 1997
Radial displacements of an infinite liquid saturated porous medium with spherical cavity
The components of radial displacements in solid and liquid parts of a liquidsaturated porous medium with spherical cavity subjected to an arbitrary time dependent force have been obtained. Laplace transform technique has been used to solve the problem. Numerical calculation has been performed for two specific models. Variations of displacement in solid and liquid parts of the medium have been shown graphically.
pp 71-87 February 1997
Steady-state thermal stresses in an infinite elastic medium containing an annular crack
Rina Bhowmick Bikash Ranjan Das
An axisymmetric steady-state thermoelastic problem of an infinite isotropic medium containing an annular crack is considered. The faces of the crack are exposed to prescribed temperature distribution. The normal components of stress and displacement on the crack plane and the stress-intensity factors at the boundaries of the crack are expressed in power series in terms of the ratio between the radii of the inner and outer boundaries. These are illustrated graphically.
pp 89-93 February 1997
On the incoming water waves against a vertical cliff
The problems of obliquely incident surface water waves against a vertical cliff have been handled in both the cases of water of infinite as well as finite depth by straight-forward uses of appropriate Havelock-type expansion theorems. The logarithmic singularity along the shore-line has been incorporated in a direct manner, by suitably representing the Dirac’s delta function.
pp 95-100 February 1997
Discrete fourier transform computation using prime Ramanujan numbers
Ramanujan numbers were introduced in [2] to implement discrete fourier transform (DFT) without using any multiplication operation. Ramanujan numbers are related to π and integers which are powers of 2. If the transform sizeN, is a Ramanujan number, then the computational complexity of the algorithms used for computing isO(N^{2}) addition and shift operations, and no multiplications. In these algorithms, the transform can be computed sequentially with a single adder inO(N^{2}) addition times. Parallel implementation of the algorithm can be executed inO(N) addition times, withO(N) number of adders. Some of these Ramanujan numbers of order-2 are related to the Biblical and Babylonian values of π [1]. In this paper, we analytically obtain upper bounds on the degree of approximation in the computation of DFT if JV is a prime Ramanujan number.
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