Volume 109, Issue 4
November 1999, pages 333-456
pp 333-343 November 1999
On hypothesis K for biquadrates
It is proved that the number of ways of expressing a large positive integern as the sum of four biquadrates is$$O\left[ {\frac{{n^{1/2} (\log \log n)^8 }}{{\log n}}} \right]$$
pp 345-351 November 1999
Rings with all modules residually finite
Define a ringA to be RRF (resp. LRF) if every right (resp. left) A-module is residually finite. Refer to A as an RF ring if it is simultaneously RRF and LRF. The present paper is devoted to the study of the structure of RRF (resp. LRF) rings. We show that all finite rings are RF. IfA is semiprimary, we show thatA is RRF ⇔A is finite ⇔A is LRF. We prove that being RRF (resp. LRF) is a Morita invariant property. All boolean rings are RF. There are other infinite strongly regular rings which are RF. IfA/J(A) is of bounded index andA does not contain any infinite family of orthogonal idempotents we prove:
A an RRF ring ⇔ A right perfect andA/J(A) finite (henceA/J(A) finite semisimple artinian).
A an LRF ring ⇔ A left perfect andA/J(A) finite
IfA is one sided quasi-duo (left or right immaterial) not containing any infinite family of orthogonal idempotents then (i) and (ii) are valid with the further strengthening thatA/J(A) is a finite product of finite fields.
pp 353-358 November 1999
Holomorphic and algebraic vector bundles on 0-convex algebraic surfaces
LetX be a smooth complex algebraic surface such that there is a proper birational morphism/:X → Y withY an affine variety. Let X_{hol} be the 2-dimensional complex manifold associated toX. Here we give conditions onX which imply that every holomorphic vector bundle onX is algebraizable and it is an extension of line bundles. We also give an approximation theorem of holomorphic vector bundles on X_{hol} (X normal algebraic surface) by algebraic vector bundles.
pp 359-372 November 1999
Means, homomorphisms, and compactifications of weighted semitopological semigroups
A A Khadem-Maboudi M A Pourabdollah
We consider some almost periodic type function algebras on a weighted semitopological semigroup, and using the set of multiplicative means on each of these algebras, we define their corresponding weighted semigroup compactifications. This will constitute an effective tool for investigating the properties of the function algebras concerned. We also show that these compactifications do not retain all the nice properties of the ordinary semigroup compactifications unless we impose some restrictions on the weight functions.
pp 373-378 November 1999
In this paper we show that for a bounded linear operatorA on a complex Hilbert spaceH, the points on the boundary of the numerical range ofA with infinite curvature and unique tangent are in the essential spectrum ofA, thus positively answering a conjecture raised by Hubner in [3].
pp 379-383 November 1999
In this note we prove a relation between the Riemann Zeta function, ζ and the ξ function (Krein spectral shift) associated with the harmonic oscillator in one dimension. This gives a new integral representation of the zeta function and also a reformulation of the Riemann hypothesis as a question inL^{1}(ℝ).
pp 385-388 November 1999
Inverse theory of Schrödinger matrices
In this note we discuss the inverse spectral theory for Schrödinger matrices, in particular a conjecture of Gesztesy-Simon [1] on the number of distinct iso-spectral Schrödinger matrices. We consider 3 × 3 matrices and obtain counter examples to their conjecture.
pp 389-395 November 1999
A nonstandard definition of finite order ultradistributions
Taking into account that finite-order ultradistributions are inverse Fourier transforms of finite-order distributions a nonstandard representation is obtained for one-dimensional finite-order ultradistributions.
pp 397-400 November 1999
Moments of escape times of random walk
Extending an idea of Spitzer [2], a way to compute the moments of the time of escape from (−N,L) by a symmetric simple random walk is exhibited. It is shown that all these moments depend polynomially onL andN.
pp 401-409 November 1999
Explicit generalized solutions to a system of conservation laws
This paper studies a special 3 by 3 system of conservation laws which cannot be solved in the classical distributional sense. By adding a viscosity term and writing the system in the form of a matrix Burgers equation an explicit formula is found for the solution of the pure initial value problem. These regularized solutions are used to construct solutions for the conservation laws with initial conditions, in the algebra of generalized functions of Colombeau. Special cases of this system were studied previously by many authors.
pp 411-416 November 1999
Boundary stabilization of a hybrid Euler—Bernoulli beam
We consider a problem of boundary stabilization of small flexural vibrations of a flexible structure modeled by an Euler-Bernoulli beam which is held by a rigid hub at one end and totally free at the other. The hub dynamics leads to a hybrid system of equations. By incorporating a condition of small rate of change of the deflection with respect tox as well ast, over the length of the beam, for appropriate initial conditions, uniform exponential decay of energy is established when a viscous boundary damping is present at the hub end.
pp 417-423 November 1999
A note on the dispersion of Love waves in layered monoclinic elastic media
Sarva Jit Singh Neelam Sachdeva Sandhya Khurana
The dispersion equation for Love waves in a monoclinic elastic layer of uniform thickness overlying a monoclinic elastic half-space is derived by applying the traction-free boundary condition at the surface and continuity conditions at the interface. The dispersion curves showing the effect of anisotropy on the calculated phase velocity are presented. The special cases of orthotropic and transversely isotropic media are also considered. It is shown that the well-known dispersion equation for Love waves in an isotropic layer overlying an isotropic half-space follows as a particular case.
pp 425-433 November 1999
Elastic wave propagation in a cylindrical bore situated in a micropolar elastic medium with stretch
Propagation of surface elastic waves in a cylindrical bore through a micropolar elastic medium with stretch is analysed in two cases. In the first case, the cylindrical bore is considered empty while in the second case, the bore is filled with homogeneous inviscid liquid. In both the problems, period equations are obtained in closed form. The problem of Banerji and Sengupta [2,3] has been reduced as a special case. Numerical calculations have been performed for a particular model and results obtained are presented graphically. It is noticed that the effect of micropolarity on dispersion curve is significant while the effect of micro-stretch on dispersion curve is not appreciable.
pp 435-440 November 1999
Solutions of one dimensional steady flow of dusty gas in an anholonomic co-ordinate system
C S Bagewadi A N Shantharajappa
We study the geometry of one-dimensional (i.e. unidirectional) incompressible steady dusty gas flow in Frenet frame field system (anholonomic co-ordinate system) by assuming the paths of velocities of dust and fluid phases to be in the same direction. The intrinsic decompositions of the basic equation are carried out and solutions for velocity of fluid phaseu, velocity of dust phasev and pressure of the fluid are obtained in terms of spin coefficients, i.e. geometrical parameters like curvatures and torsions of the streamline when the flow is
parallel straight line i.e.k_{s} = 0
parallel andk_{s} ≠ 0, under the assumption that, the sum of the deformations at a point of the fluid surface along the stream line, its principal normal and binormal is constant
Further, we have proved a result, which is an extension of Barron, and a graph ofp againsts is plotted (figure 1).
pp 441-452 November 1999
Stokes drag on axially symmetric bodies: a new approach
Sunil Datta Deepak Kumar Srivastava
In this paper a new approach to evaluate the drag force in a simple way on a restricted axially symmetric body placed in a uniform stream (i) parallel to its axis, (ii) transverse to its axis, is advanced when the flow is governed by the Stokes equations. The method exploits the well-known integral for evaluating the drag on a sphere. The method not only provides the value of the drag on prolate and oblate spheroids and a deformed sphere in axial flow which already exists in literature but also new results for a cycloidal body, an egg shaped body and a deformed sphere in transverse flow. The salient results are exhibited graphically. The limitations imposed on the analysis because of the lack of fore and aft symmetry in the case of an eggshaped body is also indicated. It is also seen that the analysis can be extended to calculate the couple on a body rotating about its axis of symmetry.
pp 453-456 November 1999
Stability of an expanding bubble in the Rayleigh model
B B Chakraborty Dinesh Khattar
A bubble expands adiabatically in an incompressible, inviscid liquid. The variation of its radiusR with time is given by the Rayleigh’s equation. We find that the bubble is stable at the equilibrium point in this model.
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