• Volume 107, Issue 1

February 1997,   pages  1-100

• 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 function2Φ1. Their continued fractions representations have also been given.

• On degrees of maps between Grassmannians

Let Gn,k denote the oriented grassmann manifold of orientedk-planes in ℝn. It is shown that for any continuous mapf: Gn,k → Gn,k, dim Gn,k = dim Gm,l = l(m −l), the Brouwer’s degree is zero, providedl &gt; 1,n ≠ m. Similar results for continuous mapsg: ℂGm,l → ℂGn,k,h: ℍGm,l → ℍGn,k, 1 ≤ l &lt; k ≤ n/2, k(n — k) = l(m — l) are also obtained.

• 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 forSL4, 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.

• Double convolution integral equations involving a general class of multivariable polynomials and the multivariableH-functions

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.

• 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 sequencexn of unit vectors inX, ifx*n is any sequence of unit vectors inX* that attain their norm at xn’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.

• Intrinsic geometry of curves and the Bonnor’s equation

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.

• 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.

• Steady-state thermal stresses in an infinite elastic medium containing an annular crack

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.

• 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.

• 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(N2) addition and shift operations, and no multiplications. In these algorithms, the transform can be computed sequentially with a single adder inO(N2) 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.

• # Proceedings – Mathematical Sciences

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November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019