Volume 95, Issue 2
December 1986, pages 79-153
pp 79-96 December 1986
Ito’s stochastic integral is defined with respect to a Wiener process taking values in a locally convex space and Ito’s formula is proved. Existence and uniqueness theorem is proved in a locally convex space for a class of stochastic evolution equations with white noise as a stochastic forcing term. The stochastic forcing term is modelled by a locally convex space valued stochastic integral.
pp 97-107 December 1986
A probability algebra (A, *, ω) consisting of a*algebraA with a faithful state ω provides a framework for an unbounded noncommutative probability theory. A characterization of symmetric probability algebra is obtained in terms of an unbounded strong commutant of the left regular representation ofA. Existence of coarse-graining is established for states that are absolutely continuous or continuous in the induced topology. Sufficiency of a*subalgebra relative to a family of states is discussed in terms of noncommutative Radon-Nikodym derivatives (a form of Halmos-Savage theorem), and is applied to couple of examples (including the canonical algebra of one degree of freedom for Heisenberg commutation relation) to obtain unbounded analogues of sufficiency results known in probability theory over a von Neumann algebra.
pp 109-120 December 1986
Effects of torsional loading in an axisymmetric micropolar elastic half-space are studied. The components of microrotation displacement, force stress and couple stress are obtained for a half-space subjected to an arbitrary load produced by shearing stress. A special case of a particular type of twist has been discussed in detail for a specific model and the micropolar effects have been shown graphically.
pp 121-125 December 1986
A new subclass of Bazilevic functions is defined and some of its properties have been studied.
pp 127-132 December 1986
Formulae for the number of different integral solutions ofa2+b2+c2+d2+ac+bd=p are given wherep is a prime and the solution satisfies certain natural congruence conditions. Similar formulae are given for the case of the quadratic forma2+b2+2c2+2d2+ac+bd.
pp 133-140 December 1986
A general theorem unifying a novel class of bilateral generating functions of certain special functions is established. A number of applications of the theorem are also given.
pp 141-153 December 1986
Sufficient conditions for somem-th order finite difference equations are presented which have a solution behaving in a precisely specified way like a given polynomial.