• Volume 94, Issue 1

September 1985,   pages  1-60

• Composition of variable co-efficient singular integral operators

In this paper we give a formula for composition of two singular integral operators with variable co-efficients by explicitly calculating the lower order terms. Also we discuss the boundedness of the lower order terms inLp-spaces.

• The homomorphismM*⊗N → Hom (M,N)

In this note we prove two theorems. In theorem 1 we prove that if M andN are two non-zero reflexive modules of finite projective dimensions over a Gorenstein local ring, such that Hom (M, N) is a third module of syzygies, then the natural homomorphismM* ⊗N → Hom (M, N) is an isomorphism. This extends the result in . In theorem 2, we prove that projective dimension of a moduleM over a regular local ringR is less than or equal ton if and only if ExtRn (M, R) ⊗M → ExtRn (M, M) is surjective; in which case it is actually bijective. This extends the usual criterion for the projectivity of a module.

• On the propagation of a multi-dimensional shock of arbitrary strength

In this paper the kinematics of a curved shock of arbitrary strength has been discussed using the theory of generalised functions. This is the extension of Moslov’s work where he has considered isentropic flow even across the shock. The condition for a nontrivial jump in the flow variables gives the shock manifold equation (sme). An equation for the rate of change of shock strength along the shock rays (defined as the characteristics of the sme) has been obtained. This exact result is then compared with the approximate result of shock dynamics derived by Whitham. The comparison shows that the approximate equations of shock dynamics deviate considerably from the exact equations derived here. In the last section we have derived the conservation form of our shock dynamic equations. These conservation forms would be very useful in numerical computations as it would allow us to derive difference schemes for which it would not be necessary to fit the shock-shock explicitly.

• Distance characterizations of finite lattices

Semimodular, modular and distributive finite lattices are characterized by means of convex sublattices and distance closed sets of the Hasse diagram graphs.

• Closed set characterizations of finite lattices

Semimodularity and the Jordan-Hölder chain condition are characterized in a finite latticeL by means of special closed sets ofL.

• Slow magnetohydrodynamic flow past a circular cylinder

Oseen’ approximations are used to study the slow motion of a viscous, incompressible, electrically conducting fluid past a circular cylinder in the presence of a uniform aligned magnetic field. Using series truncation method, the analytical solutions for the first three terms in the Fourier sine series expansion of the stream function are obtained. Numerical values of the tangential drag for different values of magnetic interaction parameter and viscous Reynolds number are calculated.